The representation theory of the restricted rational Cherednik algebra for G14

Computed by Ulrich Thiel using CHAMP (see LMS J. Comput. Math., 2015). Last update on Fri Mar 27 12:48:17 CET 2015.

Note: In the larger tables each cell has a mouseover tooltip providing information about the cell.

Quick navigation: Exceptional hyperplanes

For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1
ϕ1,12 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1
ϕ1,28 1 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 2 2 2 2 2 2 2 0 0 1 0 1 0 0 2 1 0 0 2 1 0 2 2 2
ϕ2,12 2 2 2 2 2 2 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 2 2 2
ϕ2,9 2 2 2 2 2 2 0 0 2 1 0 1 2 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 2 2 2 2 2 2 2 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 2 2
ϕ2,8 2 2 2 2 2 2 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 2 2 2
ϕ2,5 2 2 2 2 2 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 2 2 2 2
ϕ2,7 2 2 2 2 2 2 1 0 1 2 0 0 2 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 2 2 2 2 2 2 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 2
ϕ2,1 2 2 2 2 2 2 1 0 1 0 0 2 0 0 2 2 0 0 1 0 1 2 2 2
ϕ3,2 3 3 3 3 3 3 2 0 1 1 0 2 0 1 1 2 0 0 2 1 1 3 3 3
ϕ3,8 3 3 3 3 3 3 1 0 2 2 0 1 1 1 0 0 2 2 0 1 1 3 3 3
ϕ3,4 3 3 3 3 3 3 0 1 1 1 0 2 2 0 1 1 1 2 0 0 2 3 3 3
ϕ3,10 3 3 3 3 3 3 1 1 0 2 0 1 1 0 2 1 1 0 2 2 0 3 3 3
ϕ3,6' 3 3 3 3 3 3 2 0 1 1 1 0 2 0 1 0 2 1 1 2 0 3 3 3
ϕ3,6'' 3 3 3 3 3 3 1 0 2 0 1 1 1 0 2 2 0 1 1 0 2 3 3 3
ϕ4,3 4 4 4 4 4 4 1 1 1 2 0 2 1 1 1 1 1 2 2 1 1 4 4 4
ϕ4,5 4 4 4 4 4 4 1 1 1 1 1 1 2 0 2 1 1 1 1 2 2 4 4 4
ϕ4,7 4 4 4 4 4 4 2 0 2 1 1 1 1 1 1 2 2 1 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 t20 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 t8 t28 t12 t20 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 t20 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 t12 t20 t28 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 t12 t20 t16 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 t28 t12 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 t + t19 t15 + t21 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 t6 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 t7 + t13 t9 + t27 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 t5 + t23 t + t19 t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 t11 + t17 t7 + t13 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 t9 + t27 t5 + t23 t7 + t13 t15 + t21 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 t15 + t21 t11 + t17 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 t5 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 t + t7 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

Exceptional hyperplanes

k2,2
k2,1
k2,1 − k2,2
k1,1
k1,1 − k2,2
k1,1 + k2,2
k1,1 − 2k2,1 + k2,2
k1,1 − k2,1
k1,1 − k2,1 − k2,2
k1,1 − k2,1 + k2,2
k1,1 − k2,1 + 2k2,2
k1,1 + k2,1
k1,1 + k2,1 − 2k2,2
k1,1 + k2,1 − k2,2
k1,1 + k2,1 + k2,2
k1,1 + 2k2,1 − k2,2
2k1,1 − 2k2,1 + k2,2
2k1,1 − k2,1 − k2,2
2k1,1 − k2,1 + 2k2,2
2k1,1 + k2,1 − 2k2,2
2k1,1 + k2,1 + k2,2
2k1,1 + 2k2,1 − k2,2

For the generic point of the hyperplane k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ4,3,  ϕ4,5},   {ϕ1,12,  ϕ1,20},   {ϕ1,0,  ϕ1,8},   {ϕ2,8,  ϕ2,5,  ϕ4,7,  ϕ2,15,  ϕ2,12,  ϕ2,9,  ϕ2,11},   {ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,09611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ1,84811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ1,204811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,15822 + 4t + 2t2
ϕ2,121642 + 4t + 6t2 + 4t3
ϕ2,9822 + 4t + 2t2
ϕ2,113022 + 4t + 6t2 + 6t3 + 6t4 + 4t5 + 2t6
ϕ2,8242
ϕ2,53022 + 4t + 6t2 + 6t3 + 6t4 + 4t5 + 2t6
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,39644 + 8t + 12t2 + 16t3 + 16t4 + 16t5 + 12t6 + 8t7 + 4t8
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,71684 + 6t + 4t2 + 2t3

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0
ϕ1,8 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0
ϕ1,16 1 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0
ϕ1,12 1 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0
ϕ1,20 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0
ϕ1,28 1 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0
ϕ2,15 1 1 2 1 1 2 1 0 0 1 0 0 0 0 2 1 0 0 2 1 0 1 1 0
ϕ2,12 1 1 2 1 1 2 0 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0
ϕ2,9 1 1 2 1 1 2 0 0 1 0 0 1 2 0 0 0 1 2 0 0 1 1 1 0
ϕ2,11 2 0 2 2 0 2 0 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 0 0
ϕ2,8 2 0 2 2 0 2 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 2 0 0
ϕ2,5 2 0 2 2 0 2 0 0 0 0 0 2 1 0 1 1 0 1 0 0 2 2 0 0
ϕ2,7 1 1 2 1 1 2 0 0 1 1 0 0 2 0 0 0 2 1 0 1 0 1 1 0
ϕ2,4 1 1 2 1 1 2 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1
ϕ2,1 1 1 2 1 1 2 1 0 0 0 0 1 0 0 2 2 0 0 1 0 1 1 1 0
ϕ3,2 2 1 3 2 1 3 1 0 0 1 0 1 0 1 1 2 0 0 2 1 1 2 1 0
ϕ3,8 2 1 3 2 1 3 0 0 1 1 0 1 1 1 0 0 2 2 0 1 1 2 1 0
ϕ3,4 2 1 3 2 1 3 0 1 0 0 0 1 2 0 1 1 1 2 0 0 2 2 1 0
ϕ3,10 2 1 3 2 1 3 0 1 0 1 0 0 1 0 2 1 1 0 2 2 0 2 1 0
ϕ3,6' 2 1 3 2 1 3 0 0 0 1 0 0 2 0 1 0 2 1 1 2 0 2 1 1
ϕ3,6'' 2 1 3 2 1 3 0 0 0 0 0 1 1 0 2 2 0 1 1 0 2 2 1 1
ϕ4,3 3 1 4 3 1 4 0 1 0 1 0 1 1 1 1 1 1 2 2 1 1 3 1 0
ϕ4,5 2 2 4 2 2 4 0 1 0 0 0 0 2 0 2 1 1 1 1 2 2 2 2 1
ϕ4,7 3 1 4 3 1 4 0 0 0 1 0 1 1 1 1 2 2 1 1 1 1 3 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 t16 t12 0 t28 0 0 0 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 0 0
ϕ1,8 0 1 t8 0 t12 t20 0 0 t 0 0 0 t5 0 0 0 t6 t2 0 0 0 0 t3 0
ϕ1,16 t8 0 1 t20 0 t12 0 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 0 0
ϕ1,12 t12 0 t28 1 0 t16 0 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 0 0
ϕ1,20 0 t12 t20 0 1 t8 t 0 0 0 0 0 0 0 t5 t6 0 0 t2 0 0 0 t3 0
ϕ1,28 t20 0 t12 t8 0 1 0 0 0 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 0 0
ϕ2,15 t9 t11 t + t19 t15 t5 t7 + t13 1 0 0 t2 0 0 0 0 t4 + t10 t5 0 0 t + t7 t3 0 t6 t2 0
ϕ2,12 t12 t8 t4 + t16 t12 t8 t4 + t16 0 1 0 0 0 0 0 t4 0 t2 t2 0 0 t6 t6 t3 t5 0
ϕ2,9 t15 t5 t7 + t13 t9 t11 t + t19 0 0 1 0 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 t6 t2 0
ϕ2,11 t7 + t13 0 t5 + t23 t + t19 0 t11 + t17 0 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 0 0
ϕ2,8 t4 + t16 0 t8 + t20 t4 + t16 0 t8 + t20 0 0 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 0 0
ϕ2,5 t + t19 0 t11 + t17 t7 + t13 0 t5 + t23 0 0 0 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 0 0
ϕ2,7 t11 t t9 + t27 t5 t7 t15 + t21 0 0 t2 t4 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 t4 0
ϕ2,4 t8 t4 t12 + t24 t8 t4 t12 + t24 0 0 0 0 0 0 0 1 0 0 0 t6 t6 t2 t2 t5 t t3
ϕ2,1 t5 t7 t15 + t21 t11 t t9 + t27 t2 0 0 0 0 t4 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 t4 0
ϕ3,2 t4 + t10 t6 t2 + t14 + t20 t10 + t16 t6 t8 + t14 + t26 t 0 0 t3 0 t3 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 t + t7 t3 0
ϕ3,8 t10 + t16 t6 t8 + t14 + t26 t4 + t10 t6 t2 + t14 + t20 0 0 t t3 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + t7 t3 0
ϕ3,4 t2 + t14 t4 t6 + t12 + t18 t8 + t14 t10 t6 + t18 + t24 0 t2 0 0 0 t t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 + t8 2t5 t 0
ϕ3,10 t8 + t14 t10 t6 + t18 + t24 t2 + t14 t4 t6 + t12 + t18 0 t2 0 t 0 0 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 2t5 t 0
ϕ3,6' t6 + t12 t2 t4 + t10 + t22 t6 + t18 t8 t10 + t16 + t22 0 0 0 t5 0 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 t5 t
ϕ3,6'' t6 + t18 t8 t10 + t16 + t22 t6 + t12 t2 t4 + t10 + t22 0 0 0 0 0 t5 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 t5 t
ϕ4,3 t3 + t9 + t15 t5 t7 + t13 + t19 + t25 t3 + t9 + t15 t5 t7 + t13 + t19 + t25 0 t3 0 t2 0 t2 t4 t t4 t5 t5 t + t7 t + t7 t3 t3 1 + 2t6 t2 0
ϕ4,5 t7 + t13 t3 + t9 t5 + t11 + t17 + t23 t7 + t13 t3 + t9 t5 + t11 + t17 + t23 0 t 0 0 0 0 t2 + t8 0 t2 + t8 t3 t3 t5 t5 t + t7 t + t7 2t4 1 + t6 t2
ϕ4,7 t5 + t11 + t17 t7 t3 + t9 + t15 + t21 t5 + t11 + t17 t7 t3 + t9 + t15 + t21 0 0 0 t4 0 t4 t6 t3 t6 t + t7 t + t7 t3 t3 t5 t5 2t2 + t8 t4 1

For the generic point of the hyperplane k2,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ1,16},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ1,12,  ϕ1,28},   {ϕ4,5,  ϕ2,7,  ϕ2,4,  ϕ2,1,  ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ4,3,  ϕ4,7},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ3,8,  ϕ3,10,  ϕ3,6''}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,04811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,169611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ1,124811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,289611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ2,153022 + 4t + 6t2 + 6t3 + 6t4 + 4t5 + 2t6
ϕ2,12242
ϕ2,93022 + 4t + 6t2 + 6t3 + 6t4 + 4t5 + 2t6
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,7822 + 4t + 2t2
ϕ2,41642 + 4t + 6t2 + 4t3
ϕ2,1822 + 4t + 2t2
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,51684 + 6t + 4t2 + 2t3
ϕ4,79644 + 8t + 12t2 + 16t3 + 16t4 + 16t5 + 12t6 + 8t7 + 4t8

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 0 1 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0
ϕ1,8 0 1 1 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1
ϕ1,16 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1
ϕ1,12 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0
ϕ1,20 0 1 1 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1
ϕ1,28 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1
ϕ2,15 0 2 2 0 2 2 2 0 0 1 0 1 0 0 0 1 0 0 2 1 0 0 0 2
ϕ2,12 0 2 2 0 2 2 0 1 0 1 0 1 0 0 0 1 1 0 0 1 1 0 0 2
ϕ2,9 0 2 2 0 2 2 0 0 2 1 0 1 0 0 0 0 1 2 0 0 1 0 0 2
ϕ2,11 1 2 1 1 2 1 1 0 0 2 0 0 1 0 0 0 1 0 1 2 0 1 0 1
ϕ2,8 1 2 1 1 2 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 1 1
ϕ2,5 1 2 1 1 2 1 0 0 1 0 0 2 0 0 1 1 0 1 0 0 2 1 0 1
ϕ2,7 1 2 1 1 2 1 0 0 1 2 0 0 1 0 0 0 2 1 0 1 0 1 0 1
ϕ2,4 1 2 1 1 2 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 0 1
ϕ2,1 1 2 1 1 2 1 1 0 0 0 0 2 0 0 1 2 0 0 1 0 1 1 0 1
ϕ3,2 1 3 2 1 3 2 1 0 0 1 0 2 0 1 0 2 0 0 2 1 1 1 0 2
ϕ3,8 1 3 2 1 3 2 0 0 1 2 0 1 0 1 0 0 2 2 0 1 1 1 0 2
ϕ3,4 1 3 2 1 3 2 0 0 1 1 0 2 0 0 0 1 1 2 0 0 2 1 1 2
ϕ3,10 1 3 2 1 3 2 1 0 0 2 0 1 0 0 0 1 1 0 2 2 0 1 1 2
ϕ3,6' 1 3 2 1 3 2 1 0 1 1 1 0 1 0 0 0 2 1 1 2 0 1 0 2
ϕ3,6'' 1 3 2 1 3 2 1 0 1 0 1 1 0 0 1 2 0 1 1 0 2 1 0 2
ϕ4,3 2 4 2 2 4 2 0 0 0 2 0 2 0 1 0 1 1 2 2 1 1 2 1 2
ϕ4,5 1 4 3 1 4 3 1 0 1 1 1 1 0 0 0 1 1 1 1 2 2 1 1 3
ϕ4,7 1 4 3 1 4 3 1 0 1 1 1 1 0 1 0 2 2 1 1 1 1 1 0 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 0 t12 t20 0 0 0 0 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 0 0
ϕ1,8 0 1 t8 0 t12 t20 0 0 t t9 0 0 0 0 0 0 t6 t2 0 0 0 0 0 t5
ϕ1,16 0 t16 1 0 t28 t12 t5 0 0 t 0 0 0 0 0 0 0 0 t6 t2 0 0 0 t3
ϕ1,12 t12 t20 0 1 t8 0 0 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 0 0
ϕ1,20 0 t12 t20 0 1 t8 t 0 0 0 0 t9 0 0 0 t6 0 0 t2 0 0 0 0 t5
ϕ1,28 0 t28 t12 0 t16 1 0 0 t5 0 0 t 0 0 0 0 0 t6 0 0 t2 0 0 t3
ϕ2,15 0 t11 + t17 t + t19 0 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 0 t5 0 0 t + t7 t3 0 0 0 2t4
ϕ2,12 0 t8 + t20 t4 + t16 0 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 0 0 t2 t2 0 0 t6 t6 0 0 t + t7
ϕ2,9 0 t5 + t23 t7 + t13 0 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 0 0 0 0 t5 t + t7 0 0 t3 0 0 2t4
ϕ2,11 t7 t15 + t21 t5 t t9 + t27 t11 t4 0 0 1 + t6 0 0 t2 0 0 0 t3 0 t5 t + t7 0 t4 0 t2
ϕ2,8 t4 t12 + t24 t8 t4 t12 + t24 t8 0 0 0 0 1 0 0 0 0 t6 t6 t2 t2 0 0 t t3 t5
ϕ2,5 t t9 + t27 t11 t7 t15 + t21 t5 0 0 t4 0 0 1 + t6 0 0 t2 t3 0 t5 0 0 t + t7 t4 0 t2
ϕ2,7 t11 t + t19 t9 t5 t7 + t13 t15 0 0 t2 t4 + t10 0 0 1 0 0 0 t + t7 t3 0 t5 0 t2 0 t6
ϕ2,4 t8 t4 + t16 t12 t8 t4 + t16 t12 0 0 0 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 t5 0 t3
ϕ2,1 t5 t7 + t13 t15 t11 t + t19 t9 t2 0 0 0 0 t4 + t10 0 0 1 t + t7 0 0 t3 0 t5 t2 0 t6
ϕ3,2 t4 t6 + t12 + t18 t2 + t14 t10 t6 + t18 + t24 t8 + t14 t 0 0 t3 0 t3 + t9 0 t2 0 1 + t6 0 0 t2 + t8 t4 t4 t 0 2t5
ϕ3,8 t10 t6 + t18 + t24 t8 + t14 t4 t6 + t12 + t18 t2 + t14 0 0 t t3 + t9 0 t3 0 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t 0 2t5
ϕ3,4 t2 t4 + t10 + t22 t6 + t12 t8 t10 + t16 + t22 t6 + t18 0 0 t5 t7 0 t + t7 0 0 0 t4 t4 1 + t6 0 0 t2 + t8 t5 t 2t3
ϕ3,10 t8 t10 + t16 + t22 t6 + t18 t2 t4 + t10 + t22 t6 + t12 t5 0 0 t + t7 0 t7 0 0 0 t4 t4 0 1 + t6 t2 + t8 0 t5 t 2t3
ϕ3,6' t6 t2 + t14 + t20 t4 + t10 t6 t8 + t14 + t26 t10 + t16 t3 0 t3 t5 t2 0 t 0 0 0 t2 + t8 t4 t4 1 + t6 0 t3 0 t + t7
ϕ3,6'' t6 t8 + t14 + t26 t10 + t16 t6 t2 + t14 + t20 t4 + t10 t3 0 t3 0 t2 t5 0 0 t t2 + t8 0 t4 t4 0 1 + t6 t3 0 t + t7
ϕ4,3 t3 + t9 t5 + t11 + t17 + t23 t7 + t13 t3 + t9 t5 + t11 + t17 + t23 t7 + t13 0 0 0 t2 + t8 0 t2 + t8 0 t 0 t5 t5 t + t7 t + t7 t3 t3 1 + t6 t2 2t4
ϕ4,5 t7 t3 + t9 + t15 + t21 t5 + t11 + t17 t7 t3 + t9 + t15 + t21 t5 + t11 + t17 t4 0 t4 t6 t3 t6 0 0 0 t3 t3 t5 t5 t + t7 t + t7 t4 1 2t2 + t8
ϕ4,7 t5 t7 + t13 + t19 + t25 t3 + t9 + t15 t5 t7 + t13 + t19 + t25 t3 + t9 + t15 t2 0 t2 t4 t t4 0 t3 0 t + t7 t + t7 t3 t3 t5 t5 t2 0 1 + 2t6

For the generic point of the hyperplane k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,8,  ϕ4,3,  ϕ2,5,  ϕ2,7,  ϕ2,4,  ϕ2,1,  ϕ2,11},   {ϕ4,5,  ϕ4,7},   {ϕ1,8,  ϕ1,16},   {ϕ1,20,  ϕ1,28},   {ϕ3,2,  ϕ3,4,  ϕ3,6'}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,89611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ1,164811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,209611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ1,284811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,11822 + 4t + 2t2
ϕ2,81642 + 4t + 6t2 + 4t3
ϕ2,5822 + 4t + 2t2
ϕ2,73022 + 4t + 6t2 + 6t3 + 6t4 + 4t5 + 2t6
ϕ2,4242
ϕ2,13022 + 4t + 6t2 + 6t3 + 6t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,31684 + 6t + 4t2 + 2t3
ϕ4,59644 + 8t + 12t2 + 16t3 + 16t4 + 16t5 + 12t6 + 8t7 + 4t8
ϕ4,74844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0
ϕ1,8 1 1 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0
ϕ1,16 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1
ϕ1,12 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0
ϕ1,20 1 1 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0
ϕ1,28 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1
ϕ2,15 2 1 1 2 1 1 2 0 0 1 0 0 0 0 1 1 0 0 2 1 0 0 1 1
ϕ2,12 2 1 1 2 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1
ϕ2,9 2 1 1 2 1 1 0 0 2 0 0 1 1 0 0 0 1 2 0 0 1 0 1 1
ϕ2,11 2 1 1 2 1 1 2 0 0 1 0 0 1 0 0 0 1 0 1 2 0 0 1 1
ϕ2,8 2 1 1 2 1 1 0 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 1
ϕ2,5 2 1 1 2 1 1 0 0 2 0 0 1 0 0 1 1 0 1 0 0 2 0 1 1
ϕ2,7 2 2 0 2 2 0 1 0 1 0 0 0 2 0 0 0 2 1 0 1 0 0 2 0
ϕ2,4 2 2 0 2 2 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 1 0 2 0
ϕ2,1 2 2 0 2 2 0 1 0 1 0 0 0 0 0 2 2 0 0 1 0 1 0 2 0
ϕ3,2 3 2 1 3 2 1 2 0 1 0 0 0 0 0 1 2 0 0 2 1 1 1 2 1
ϕ3,8 3 2 1 3 2 1 1 0 2 0 0 0 1 0 0 0 2 2 0 1 1 1 2 1
ϕ3,4 3 2 1 3 2 1 0 1 1 0 0 1 1 0 1 1 1 2 0 0 2 0 2 1
ϕ3,10 3 2 1 3 2 1 1 1 0 1 0 0 1 0 1 1 1 0 2 2 0 0 2 1
ϕ3,6' 3 2 1 3 2 1 2 0 1 0 1 0 1 0 0 0 2 1 1 2 0 0 2 1
ϕ3,6'' 3 2 1 3 2 1 1 0 2 0 1 0 0 0 1 2 0 1 1 0 2 0 2 1
ϕ4,3 4 3 1 4 3 1 1 1 1 0 0 0 1 0 1 1 1 2 2 1 1 1 3 1
ϕ4,5 4 3 1 4 3 1 1 1 1 0 1 0 1 0 1 1 1 1 1 2 2 0 3 1
ϕ4,7 4 2 2 4 2 2 2 0 2 0 1 0 0 0 0 2 2 1 1 1 1 1 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 0 t12 t20 0 0 0 t9 0 0 0 0 0 t t2 0 0 0 0 t6 0 t5 0
ϕ1,8 t16 1 0 t28 t12 0 0 0 t 0 0 0 t5 0 0 0 t6 t2 0 0 0 0 t3 0
ϕ1,16 t8 0 1 t20 0 t12 t5 0 0 t 0 0 0 0 0 0 0 0 t6 t2 0 0 0 t3
ϕ1,12 t12 t20 0 1 t8 0 t9 0 0 0 0 0 t 0 0 0 t2 0 0 t6 0 0 t5 0
ϕ1,20 t28 t12 0 t16 1 0 t 0 0 0 0 0 0 0 t5 t6 0 0 t2 0 0 0 t3 0
ϕ1,28 t20 0 t12 t8 0 1 0 0 t5 0 0 t 0 0 0 0 0 t6 0 0 t2 0 0 t3
ϕ2,15 t9 + t27 t11 t t15 + t21 t5 t7 1 + t6 0 0 t2 0 0 0 0 t4 t5 0 0 t + t7 t3 0 0 t2 t4
ϕ2,12 t12 + t24 t8 t4 t12 + t24 t8 t4 0 1 0 0 0 0 0 0 0 t2 t2 0 0 t6 t6 t3 t5 t
ϕ2,9 t15 + t21 t5 t7 t9 + t27 t11 t 0 0 1 + t6 0 0 t2 t4 0 0 0 t5 t + t7 0 0 t3 0 t2 t4
ϕ2,11 t7 + t13 t15 t5 t + t19 t9 t11 t4 + t10 0 0 1 0 0 t2 0 0 0 t3 0 t5 t + t7 0 0 t6 t2
ϕ2,8 t4 + t16 t12 t8 t4 + t16 t12 t8 0 t4 0 0 1 0 0 0 0 t6 t6 t2 t2 0 0 0 t3 t5
ϕ2,5 t + t19 t9 t11 t7 + t13 t15 t5 0 0 t4 + t10 0 0 1 0 0 t2 t3 0 t5 0 0 t + t7 0 t6 t2
ϕ2,7 t11 + t17 t + t19 0 t5 + t23 t7 + t13 0 t8 0 t2 0 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 0 2t4 0
ϕ2,4 t8 + t20 t4 + t16 0 t8 + t20 t4 + t16 0 t5 0 t5 0 0 0 0 1 0 0 0 t6 t6 t2 t2 0 t + t7 0
ϕ2,1 t5 + t23 t7 + t13 0 t11 + t17 t + t19 0 t2 0 t8 0 0 0 0 0 1 + t6 t + t7 0 0 t3 0 t5 0 2t4 0
ϕ3,2 t4 + t10 + t22 t6 + t12 t2 t10 + t16 + t22 t6 + t18 t8 t + t7 0 t7 0 0 0 0 0 t5 1 + t6 0 0 t2 + t8 t4 t4 t 2t3 t5
ϕ3,8 t10 + t16 + t22 t6 + t18 t8 t4 + t10 + t22 t6 + t12 t2 t7 0 t + t7 0 0 0 t5 0 0 0 1 + t6 t2 + t8 0 t4 t4 t 2t3 t5
ϕ3,4 t2 + t14 + t20 t4 + t10 t6 t8 + t14 + t26 t10 + t16 t6 0 t2 t5 0 0 t t3 0 t3 t4 t4 1 + t6 0 0 t2 + t8 0 t + t7 t3
ϕ3,10 t8 + t14 + t26 t10 + t16 t6 t2 + t14 + t20 t4 + t10 t6 t5 t2 0 t 0 0 t3 0 t3 t4 t4 0 1 + t6 t2 + t8 0 0 t + t7 t3
ϕ3,6' t6 + t12 + t18 t2 + t14 t4 t6 + t18 + t24 t8 + t14 t10 t3 + t9 0 t3 0 t2 0 t 0 0 0 t2 + t8 t4 t4 1 + t6 0 0 2t5 t
ϕ3,6'' t6 + t18 + t24 t8 + t14 t10 t6 + t12 + t18 t2 + t14 t4 t3 0 t3 + t9 0 t2 0 0 0 t t2 + t8 0 t4 t4 0 1 + t6 0 2t5 t
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 t7 t3 + t9 + t15 + t21 t5 + t11 + t17 t7 t6 t3 t6 0 0 0 t4 0 t4 t5 t5 t + t7 t + t7 t3 t3 1 2t2 + t8 t4
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 t5 t7 + t13 + t19 + t25 t3 + t9 + t15 t5 t4 t t4 0 t3 0 t2 0 t2 t3 t3 t5 t5 t + t7 t + t7 0 1 + 2t6 t2
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 t3 + t9 t5 + t11 + t17 + t23 t7 + t13 t3 + t9 t2 + t8 0 t2 + t8 0 t 0 0 0 0 t + t7 t + t7 t3 t3 t5 t5 t2 2t4 1 + t6

For the generic point of the hyperplane k1,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,8,  ϕ1,20},   {ϕ3,2,  ϕ3,8,  ϕ3,4,  ϕ3,10,  ϕ3,6',  ϕ3,6''},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ1,0,  ϕ1,12},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ1,16,  ϕ1,28}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,07211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,87211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,167211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,127211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,207211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,287211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,82433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,42433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,102433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,6'2433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,6''2433 + 6t + 6t2 + 6t3 + 3t4
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1
ϕ1,8 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 1
ϕ1,16 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1
ϕ1,12 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1
ϕ1,20 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1
ϕ1,28 0 0 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1
ϕ2,15 1 1 1 1 1 1 2 0 0 1 0 1 0 0 2 0 0 0 1 1 0 2 2 2
ϕ2,12 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 0 2 2 2
ϕ2,9 1 1 1 1 1 1 0 0 2 1 0 1 2 0 0 0 0 1 0 0 1 2 2 2
ϕ2,11 1 1 1 1 1 1 2 0 0 2 0 0 1 0 1 0 1 0 0 1 0 2 2 2
ϕ2,8 1 1 1 1 1 1 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 2 2 2
ϕ2,5 1 1 1 1 1 1 0 0 2 0 0 2 1 0 1 1 0 0 0 0 1 2 2 2
ϕ2,7 1 1 1 1 1 1 1 0 1 2 0 0 2 0 0 0 1 1 0 0 0 2 2 2
ϕ2,4 1 1 1 1 1 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 2 2 2
ϕ2,1 1 1 1 1 1 1 1 0 1 0 0 2 0 0 2 1 0 0 1 0 0 2 2 2
ϕ3,2 2 2 2 1 1 1 2 0 1 1 0 2 0 1 1 1 0 0 1 1 0 3 3 3
ϕ3,8 1 1 1 2 2 2 1 0 2 2 0 1 1 1 0 0 1 1 0 0 1 3 3 3
ϕ3,4 2 2 2 1 1 1 0 1 1 1 0 2 2 0 1 1 0 1 0 0 1 3 3 3
ϕ3,10 1 1 1 2 2 2 1 1 0 2 0 1 1 0 2 0 1 0 1 1 0 3 3 3
ϕ3,6' 2 2 2 1 1 1 2 0 1 1 1 0 2 0 1 0 1 1 0 1 0 3 3 3
ϕ3,6'' 1 1 1 2 2 2 1 0 2 0 1 1 1 0 2 1 0 0 1 0 1 3 3 3
ϕ4,3 2 2 2 2 2 2 1 1 1 2 0 2 1 1 1 0 0 1 1 1 1 4 4 4
ϕ4,5 2 2 2 2 2 2 1 1 1 1 1 1 2 0 2 1 1 0 0 1 1 4 4 4
ϕ4,7 2 2 2 2 2 2 2 0 2 1 1 1 1 1 1 1 1 1 1 0 0 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 0 0 0 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 0 t3 t5 t7
ϕ1,8 t16 1 t8 0 0 0 0 0 t t9 0 0 t5 0 0 0 0 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 0 0 0 t5 0 0 t 0 0 0 0 t9 0 0 0 0 t2 0 t5 t7 t3
ϕ1,12 0 0 0 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 0 0 t3 t5 t7
ϕ1,20 0 0 0 t16 1 t8 t 0 0 0 0 t9 0 0 t5 0 0 0 t2 0 0 t7 t3 t5
ϕ1,28 0 0 0 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 0 0 0 t2 t5 t7 t3
ϕ2,15 t9 t11 t t15 t5 t7 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 0 0 0 t t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 t8 t4 t12 t8 t4 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 0 0 t3 + t9 2t5 t + t7
ϕ2,9 t15 t5 t7 t9 t11 t 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 0 t 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 t15 t5 t t9 t11 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 0 t 0 2t4 2t6 t2 + t8
ϕ2,8 t4 t12 t8 t4 t12 t8 0 t4 0 0 1 0 t5 0 t5 0 0 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t t9 t11 t7 t15 t5 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 0 0 0 t 2t4 2t6 t2 + t8
ϕ2,7 t11 t t9 t5 t7 t15 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t t3 0 0 0 t2 + t8 2t4 2t6
ϕ2,4 t8 t4 t12 t8 t4 t12 t5 0 t5 0 t4 0 0 1 0 0 0 0 0 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 t7 t15 t11 t t9 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t 0 0 t3 0 0 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 t6 + t12 t2 + t14 t10 t6 t8 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 0 0 t2 t4 0 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 t6 t8 t4 + t10 t6 + t12 t2 + t14 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 t2 0 0 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 t4 + t10 t6 + t12 t8 t10 t6 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 0 1 0 0 t2 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 t10 t6 t2 + t14 t4 + t10 t6 + t12 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 0 t4 0 1 t2 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 t2 + t14 t4 + t10 t6 t8 t10 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 t4 0 1 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 t8 t10 t6 + t12 t2 + t14 t4 + t10 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 0 0 t4 0 1 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 t5 + t11 t7 + t13 t3 + t9 t5 + t11 t7 + t13 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 0 0 t t t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 t3 + t9 t5 + t11 t7 + t13 t3 + t9 t5 + t11 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 0 0 t t 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 t7 + t13 t3 + t9 t5 + t11 t7 + t13 t3 + t9 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 t t t3 t3 0 0 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ1,8,  ϕ2,7,  ϕ2,4,  ϕ1,12,  ϕ2,1},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ3,8,  ϕ3,10,  ϕ3,6''}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,8111
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,122511 + 2t + 3t2 + 4t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,71442 + 3t + 4t2 + 3t3 + 2t4
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 1 1 1 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1
ϕ1,12 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 1 0 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1
ϕ1,28 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 2 0 2 0 2 2 2 0 0 1 0 1 0 0 2 1 0 0 2 1 0 2 2 2
ϕ2,12 2 0 2 0 2 2 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 2 2 2
ϕ2,9 2 0 2 0 2 2 0 0 2 1 0 1 1 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 2 0 2 1 2 2 2 0 0 2 0 0 0 0 1 0 1 0 1 2 0 2 2 2
ϕ2,8 2 0 2 1 2 2 0 1 0 0 1 0 0 0 1 1 1 1 1 0 0 2 2 2
ϕ2,5 2 0 2 1 2 2 0 0 2 0 0 2 0 0 1 1 0 1 0 0 2 2 2 2
ϕ2,7 2 0 2 0 2 2 1 0 1 2 0 0 1 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 2 0 2 0 2 2 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 2
ϕ2,1 2 0 2 0 2 2 1 0 1 0 0 2 0 0 2 2 0 0 1 0 1 2 2 2
ϕ3,2 3 0 3 0 3 3 2 0 1 1 0 2 0 1 1 2 0 0 2 1 1 3 3 3
ϕ3,8 3 0 3 1 3 3 1 0 2 2 0 1 0 1 0 0 2 2 0 1 1 3 3 3
ϕ3,4 3 0 3 0 3 3 0 1 1 1 0 2 1 0 1 1 1 2 0 0 2 3 3 3
ϕ3,10 3 0 3 1 3 3 1 1 0 2 0 1 0 0 2 1 1 0 2 2 0 3 3 3
ϕ3,6' 3 0 3 0 3 3 2 0 1 1 1 0 1 0 1 0 2 1 1 2 0 3 3 3
ϕ3,6'' 3 0 3 1 3 3 1 0 2 0 1 1 0 0 2 2 0 1 1 0 2 3 3 3
ϕ4,3 4 0 4 1 4 4 1 1 1 2 0 2 0 1 1 1 1 2 2 1 1 4 4 4
ϕ4,5 4 0 4 0 4 4 1 1 1 1 1 1 1 0 2 1 1 1 1 2 2 4 4 4
ϕ4,7 4 0 4 1 4 4 2 0 2 1 1 1 0 1 1 2 2 1 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 t16 0 t20 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 t8 0 t12 t20 0 0 t t9 0 0 0 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 0 1 0 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 t12 0 t28 1 t8 t16 t9 0 0 t5 0 0 0 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 0 t20 0 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 0 t12 t8 t16 1 0 0 t5 0 0 t 0 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 0 t + t19 0 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 0 t4 + t16 0 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 t6 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 0 t7 + t13 0 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 t4 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 0 t5 + t23 t t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 0 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 0 t8 + t20 t4 t12 + t24 t8 + t20 0 t4 0 0 1 0 0 0 t5 t6 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 0 t11 + t17 t7 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 0 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 0 t9 + t27 0 t7 + t13 t15 + t21 t8 0 t2 t4 + t10 0 0 1 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 0 t12 + t24 0 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 0 t15 + t21 0 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 0 t2 + t14 + t20 0 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 0 t8 + t14 + t26 t4 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 0 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 0 t6 + t12 + t18 0 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 0 t3 t4 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 0 t6 + t18 + t24 t2 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 0 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 0 t4 + t10 + t22 0 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 0 t10 + t16 + t22 t6 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 0 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 0 t t4 t5 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 0 t2 + t8 t3 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 0 t3 t6 t + t7 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ1,0,  ϕ2,7,  ϕ2,4,  ϕ2,1,  ϕ1,20},   {ϕ3,8,  ϕ3,10,  ϕ3,6''}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,02511 + 2t + 3t2 + 4t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,20111
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,11442 + 3t + 4t2 + 3t3 + 2t4
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1
ϕ1,8 0 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 1 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1
ϕ1,12 0 1 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1
ϕ1,28 0 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 0 2 2 2 0 2 2 0 0 1 0 1 0 0 1 1 0 0 2 1 0 2 2 2
ϕ2,12 0 2 2 2 0 2 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 2 2 2
ϕ2,9 0 2 2 2 0 2 0 0 2 1 0 1 2 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 1 2 2 2 0 2 2 0 0 2 0 0 1 0 0 0 1 0 1 2 0 2 2 2
ϕ2,8 1 2 2 2 0 2 0 1 0 0 1 0 1 0 0 1 1 1 1 0 0 2 2 2
ϕ2,5 1 2 2 2 0 2 0 0 2 0 0 2 1 0 0 1 0 1 0 0 2 2 2 2
ϕ2,7 0 2 2 2 0 2 1 0 1 2 0 0 2 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 0 2 2 2 0 2 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 2
ϕ2,1 0 2 2 2 0 2 1 0 1 0 0 2 0 0 1 2 0 0 1 0 1 2 2 2
ϕ3,2 1 3 3 3 0 3 2 0 1 1 0 2 0 1 0 2 0 0 2 1 1 3 3 3
ϕ3,8 0 3 3 3 0 3 1 0 2 2 0 1 1 1 0 0 2 2 0 1 1 3 3 3
ϕ3,4 1 3 3 3 0 3 0 1 1 1 0 2 2 0 0 1 1 2 0 0 2 3 3 3
ϕ3,10 0 3 3 3 0 3 1 1 0 2 0 1 1 0 1 1 1 0 2 2 0 3 3 3
ϕ3,6' 1 3 3 3 0 3 2 0 1 1 1 0 2 0 0 0 2 1 1 2 0 3 3 3
ϕ3,6'' 0 3 3 3 0 3 1 0 2 0 1 1 1 0 1 2 0 1 1 0 2 3 3 3
ϕ4,3 1 4 4 4 0 4 1 1 1 2 0 2 1 1 0 1 1 2 2 1 1 4 4 4
ϕ4,5 0 4 4 4 0 4 1 1 1 1 1 1 2 0 1 1 1 1 1 2 2 4 4 4
ϕ4,7 1 4 4 4 0 4 2 0 2 1 1 1 1 1 0 2 2 1 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 0 t28 0 0 t9 0 0 t5 0 0 0 t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 0 1 t8 t28 0 t20 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 t20 0 t12 t5 0 0 t 0 0 0 0 0 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 0 t20 t28 1 0 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 0 t12 t20 t16 1 t8 t 0 0 0 0 t9 0 0 0 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 0 t28 t12 t8 0 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 0 t11 + t17 t + t19 t15 + t21 0 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 0 t8 + t20 t4 + t16 t12 + t24 0 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 t6 t3 + t9 2t5 t + t7
ϕ2,9 0 t5 + t23 t7 + t13 t9 + t27 0 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 t15 + t21 t5 + t23 t + t19 0 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 0 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 t12 + t24 t8 + t20 t4 + t16 0 t8 + t20 0 t4 0 0 1 0 t5 0 0 t6 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t t9 + t27 t11 + t17 t7 + t13 0 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 0 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 0 t + t19 t9 + t27 t5 + t23 0 t15 + t21 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 0 t4 + t16 t12 + t24 t8 + t20 0 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 0 t7 + t13 t15 + t21 t11 + t17 0 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 0 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 0 1 + t6 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 0 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 0 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 0 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 0 t4 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 0 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 0 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 t3 0 t3 t4 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 0 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t + t7 0 0 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' 0 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 0 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t 0 t5 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 + t8 0 t2 t3 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 0 t + t7 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 − 2k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,7,  ϕ1,16,  ϕ2,4,  ϕ2,1,  ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,16311 + 2t
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,7322 + t
ϕ2,4942 + 4t + 3t2
ϕ2,13922 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ3,2963 + 4t + 2t2
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,43933 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 4t6 + 2t7
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'1263 + 6t + 3t2
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 1 1
ϕ1,8 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1
ϕ1,12 1 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1
ϕ1,20 1 1 0 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1
ϕ1,28 1 1 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 2 2 1 2 2 2 2 0 0 1 0 1 0 0 1 0 0 0 2 0 0 2 2 2
ϕ2,12 2 2 0 2 2 2 0 1 0 1 0 1 0 0 0 1 1 0 0 0 1 2 2 2
ϕ2,9 2 2 0 2 2 2 0 0 2 1 0 1 0 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 2 2 0 2 2 2 2 0 0 2 0 0 0 0 0 0 1 0 1 1 0 2 2 2
ϕ2,8 2 2 0 2 2 2 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 2 2 2
ϕ2,5 2 2 0 2 2 2 0 0 2 0 0 2 0 0 1 0 0 1 0 0 2 2 2 2
ϕ2,7 2 2 0 2 2 2 1 0 1 2 0 0 1 0 0 0 2 1 0 0 0 2 2 2
ϕ2,4 2 2 0 2 2 2 1 0 1 0 1 0 0 1 0 0 0 0 1 0 1 2 2 2
ϕ2,1 2 2 0 2 2 2 1 0 1 0 0 2 0 0 2 0 0 0 1 0 1 2 2 2
ϕ3,2 3 3 0 3 3 3 2 0 1 1 0 2 0 0 1 1 0 0 2 0 1 3 3 3
ϕ3,8 3 3 0 3 3 3 1 0 2 2 0 1 0 1 0 0 2 1 0 0 1 3 3 3
ϕ3,4 3 3 0 3 3 3 0 1 1 1 0 2 0 0 1 0 1 2 0 0 2 3 3 3
ϕ3,10 3 3 0 3 3 3 1 1 0 2 0 1 0 0 1 0 1 0 2 1 0 3 3 3
ϕ3,6' 3 3 0 3 3 3 2 0 1 1 1 0 0 0 0 0 2 1 1 1 0 3 3 3
ϕ3,6'' 3 3 0 3 3 3 1 0 2 0 1 1 0 0 2 0 0 1 1 0 2 3 3 3
ϕ4,3 4 4 0 4 4 4 1 1 1 2 0 2 0 1 1 0 1 1 2 0 1 4 4 4
ϕ4,5 4 4 0 4 4 4 1 1 1 1 1 1 0 0 1 0 1 1 1 1 2 4 4 4
ϕ4,7 4 4 0 4 4 4 2 0 2 1 1 1 0 0 1 1 2 1 1 0 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 0 t12 t20 t28 0 0 t9 0 0 t5 0 0 t 0 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 0 t28 t12 t20 0 0 t t9 0 0 0 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 t20 t28 t12 t5 0 0 t 0 0 0 0 0 0 0 0 t6 0 0 t5 t7 t3
ϕ1,12 t12 t20 0 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 0 0 t3 t5 t7
ϕ1,20 t28 t12 0 t16 1 t8 t 0 0 0 0 t9 0 0 t5 0 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 t28 0 t8 t16 1 0 0 t5 0 0 t 0 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 t t15 + t21 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 0 0 0 t + t7 0 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 0 t12 + t24 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 0 0 t2 t2 0 0 0 t6 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 0 t9 + t27 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 0 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 0 t + t19 t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 0 0 0 0 t3 0 t5 t 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 0 t4 + t16 t12 + t24 t8 + t20 0 t4 0 0 1 0 0 0 t5 0 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 0 t7 + t13 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 0 0 t2 0 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 0 t5 + t23 t7 + t13 t15 + t21 t8 0 t2 t4 + t10 0 0 1 0 0 0 t + t7 t3 0 0 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 0 t8 + t20 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 0 t6 0 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 0 t11 + t17 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 0 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 0 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 0 t5 1 0 0 t2 + t8 0 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 0 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 0 t2 0 0 1 + t6 t2 0 0 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 0 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 0 0 t3 0 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 0 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 0 0 t3 0 t4 0 1 + t6 t2 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 0 t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 0 0 0 0 t2 + t8 t4 t4 1 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 0 t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 0 0 t + t7 0 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 0 t t4 0 t5 t t + t7 0 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 0 0 t2 0 t3 t5 t5 t t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 0 0 t6 t t + t7 t3 t3 0 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 − k2,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,8,  ϕ2,5,  ϕ1,16,  ϕ1,12,  ϕ2,11},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,162511 + 2t + 3t2 + 4t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,12111
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,111442 + 3t + 4t2 + 3t3 + 2t4
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1
ϕ1,12 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 1 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1
ϕ1,28 1 1 0 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 2 2 1 0 2 2 2 0 0 0 0 1 0 0 2 1 0 0 2 1 0 2 2 2
ϕ2,12 2 2 1 0 2 2 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 2 2 2
ϕ2,9 2 2 1 0 2 2 0 0 2 0 0 1 2 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 2 2 0 0 2 2 2 0 0 1 0 0 1 0 1 0 1 0 1 2 0 2 2 2
ϕ2,8 2 2 0 0 2 2 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 2 2 2
ϕ2,5 2 2 0 0 2 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 2 2 2 2
ϕ2,7 2 2 0 0 2 2 1 0 1 1 0 0 2 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 2 2 0 0 2 2 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 2
ϕ2,1 2 2 0 0 2 2 1 0 1 0 0 2 0 0 2 2 0 0 1 0 1 2 2 2
ϕ3,2 3 3 1 0 3 3 2 0 1 0 0 2 0 1 1 2 0 0 2 1 1 3 3 3
ϕ3,8 3 3 0 0 3 3 1 0 2 1 0 1 1 1 0 0 2 2 0 1 1 3 3 3
ϕ3,4 3 3 1 0 3 3 0 1 1 0 0 2 2 0 1 1 1 2 0 0 2 3 3 3
ϕ3,10 3 3 0 0 3 3 1 1 0 1 0 1 1 0 2 1 1 0 2 2 0 3 3 3
ϕ3,6' 3 3 1 0 3 3 2 0 1 0 1 0 2 0 1 0 2 1 1 2 0 3 3 3
ϕ3,6'' 3 3 0 0 3 3 1 0 2 0 1 1 1 0 2 2 0 1 1 0 2 3 3 3
ϕ4,3 4 4 0 0 4 4 1 1 1 1 0 2 1 1 1 1 1 2 2 1 1 4 4 4
ϕ4,5 4 4 1 0 4 4 1 1 1 0 1 1 2 0 2 1 1 1 1 2 2 4 4 4
ϕ4,7 4 4 1 0 4 4 2 0 2 0 1 1 1 1 1 2 2 1 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 0 0 t20 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 t8 0 t12 t20 0 0 t 0 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 0 t28 t12 t5 0 0 0 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 t12 t20 0 1 t8 t16 t9 0 0 0 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 t12 0 0 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 t28 0 0 t16 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 t 0 t5 + t23 t7 + t13 1 + t6 0 0 0 0 t8 0 0 t4 + t10 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 t4 0 t8 + t20 t4 + t16 0 1 0 0 0 t5 0 t4 0 t2 t2 0 0 t6 t6 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 t7 0 t11 + t17 t + t19 0 0 1 + t6 0 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 0 0 t9 + t27 t11 + t17 t4 + t10 0 0 1 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 0 0 t12 + t24 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 0 0 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 0 0 t7 + t13 t15 + t21 t8 0 t2 t4 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 0 0 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 0 0 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 0 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 0 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 0 0 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 0 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 0 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 0 0 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t 0 t7 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 0 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 0 t2 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 0 0 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 0 t2 + t8 t4 t t4 t5 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 0 t3 t6 t2 + t8 0 t2 + t8 t3 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 0 t t4 t6 t3 t6 t + t7 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 − k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ1,12,  ϕ3,8,  ϕ2,15,  ϕ2,12,  ϕ3,10,  ϕ2,9,  ϕ3,6''}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,12311 + 2t
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,153922 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ2,12942 + 4t + 3t2
ϕ2,9322 + t
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,81263 + 6t + 3t2
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,10963 + 4t + 2t2
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''3933 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 4t6 + 2t7
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 1 1 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 1
ϕ1,16 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1
ϕ1,12 1 1 1 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1
ϕ1,20 1 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1
ϕ1,28 1 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 2 2 2 0 2 2 2 0 0 1 0 1 0 0 2 1 0 0 0 1 0 2 2 2
ϕ2,12 2 2 2 0 2 2 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 2 2 2
ϕ2,9 2 2 2 0 2 2 0 0 1 1 0 1 2 0 0 0 0 2 0 0 1 2 2 2
ϕ2,11 2 2 2 1 2 2 1 0 0 2 0 0 1 0 1 0 0 0 0 2 0 2 2 2
ϕ2,8 2 2 2 0 2 2 0 0 0 0 1 0 1 0 1 1 0 1 1 0 0 2 2 2
ϕ2,5 2 2 2 0 2 2 0 0 0 0 0 2 1 0 1 1 0 1 0 0 2 2 2 2
ϕ2,7 2 2 2 0 2 2 0 0 0 2 0 0 2 0 0 0 1 1 0 1 0 2 2 2
ϕ2,4 2 2 2 0 2 2 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1 2 2 2
ϕ2,1 2 2 2 0 2 2 1 0 0 0 0 2 0 0 2 2 0 0 0 0 1 2 2 2
ϕ3,2 3 3 3 0 3 3 2 0 0 1 0 2 0 1 1 2 0 0 0 1 1 3 3 3
ϕ3,8 3 3 3 0 3 3 0 0 0 2 0 1 1 1 0 0 1 2 0 1 1 3 3 3
ϕ3,4 3 3 3 0 3 3 0 1 0 1 0 2 2 0 1 1 0 2 0 0 1 3 3 3
ϕ3,10 3 3 3 0 3 3 1 0 0 2 0 1 1 0 2 1 0 0 1 2 0 3 3 3
ϕ3,6' 3 3 3 0 3 3 1 0 0 1 1 0 2 0 1 0 1 1 0 2 0 3 3 3
ϕ3,6'' 3 3 3 0 3 3 1 0 0 0 1 1 1 0 2 2 0 1 0 0 2 3 3 3
ϕ4,3 4 4 4 0 4 4 1 0 0 2 0 2 1 1 1 1 0 2 1 1 1 4 4 4
ϕ4,5 4 4 4 0 4 4 1 1 0 1 1 1 2 0 2 1 0 1 0 2 1 4 4 4
ϕ4,7 4 4 4 0 4 4 1 0 0 1 1 1 1 1 1 2 1 1 0 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 0 t20 t28 0 0 0 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 t8 0 t12 t20 0 0 t t9 0 0 t5 0 0 0 0 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 0 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 0 t2 0 t5 t7 t3
ϕ1,12 t12 t20 t28 1 t8 t16 0 0 0 t5 0 0 t 0 0 0 0 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 t12 t20 0 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 0 0 0 t7 t3 t5
ϕ1,28 t20 t28 t12 0 t16 1 0 0 0 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 t + t19 0 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 0 t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 t4 + t16 0 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 0 0 0 t6 0 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 t7 + t13 0 t11 + t17 t + t19 0 0 1 t8 0 t2 t4 + t10 0 0 0 0 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 t5 + t23 t t9 + t27 t11 + t17 t4 0 0 1 + t6 0 0 t2 0 t8 0 0 0 0 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 t8 + t20 0 t12 + t24 t8 + t20 0 0 0 0 1 0 t5 0 t5 t6 0 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 t11 + t17 0 t15 + t21 t5 + t23 0 0 0 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 t9 + t27 0 t7 + t13 t15 + t21 0 0 0 t4 + t10 0 0 1 + t6 0 0 0 t t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 t12 + t24 0 t4 + t16 t12 + t24 t5 0 0 0 t4 0 0 1 0 0 0 t6 0 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 t15 + t21 0 t + t19 t9 + t27 t2 0 0 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 0 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 0 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 0 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 0 t6 + t12 + t18 t2 + t14 + t20 0 0 0 t3 + t9 0 t3 t5 t2 0 0 1 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 0 t10 + t16 + t22 t6 + t18 + t24 0 t2 0 t7 0 t + t7 t3 + t9 0 t3 t4 0 1 + t6 0 0 t2 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 t4 + t10 + t22 t6 + t12 + t18 t5 0 0 t + t7 0 t7 t3 0 t3 + t9 t4 0 0 1 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 0 t8 + t14 + t26 t10 + t16 + t22 t3 0 0 t5 t2 0 t + t7 0 t7 0 t2 t4 0 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 0 t2 + t14 + t20 t4 + t10 + t22 t3 0 0 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 0 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 0 0 t2 + t8 0 t2 + t8 t4 t t4 t5 0 t + t7 t t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t 0 t6 t3 t6 t2 + t8 0 t2 + t8 t3 0 t5 0 t + t7 t 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 0 0 t4 t t4 t6 t3 t6 t + t7 t t3 0 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 − k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ1,16,  ϕ1,20,  ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,16111
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,202511 + 2t + 3t2 + 4t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,151442 + 3t + 4t2 + 3t3 + 2t4
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 1 1 0 1 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1
ϕ1,12 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 1 1 0 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1
ϕ1,28 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 2 2 0 2 0 2 1 0 0 1 0 1 0 0 2 1 0 0 2 1 0 2 2 2
ϕ2,12 2 2 0 2 0 2 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 2 2 2
ϕ2,9 2 2 0 2 0 2 0 0 2 1 0 1 2 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 2 2 0 2 0 2 1 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 2 2
ϕ2,8 2 2 0 2 0 2 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 2 2 2
ϕ2,5 2 2 0 2 0 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 2 2 2 2
ϕ2,7 2 2 0 2 1 2 0 0 1 2 0 0 2 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 2 2 0 2 1 2 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 2
ϕ2,1 2 2 0 2 1 2 0 0 1 0 0 2 0 0 2 2 0 0 1 0 1 2 2 2
ϕ3,2 3 3 0 3 0 3 1 0 1 1 0 2 0 1 1 2 0 0 2 1 1 3 3 3
ϕ3,8 3 3 0 3 1 3 0 0 2 2 0 1 1 1 0 0 2 2 0 1 1 3 3 3
ϕ3,4 3 3 0 3 0 3 0 1 1 1 0 2 2 0 1 1 1 2 0 0 2 3 3 3
ϕ3,10 3 3 0 3 1 3 0 1 0 2 0 1 1 0 2 1 1 0 2 2 0 3 3 3
ϕ3,6' 3 3 0 3 0 3 1 0 1 1 1 0 2 0 1 0 2 1 1 2 0 3 3 3
ϕ3,6'' 3 3 0 3 1 3 0 0 2 0 1 1 1 0 2 2 0 1 1 0 2 3 3 3
ϕ4,3 4 4 0 4 1 4 0 1 1 2 0 2 1 1 1 1 1 2 2 1 1 4 4 4
ϕ4,5 4 4 0 4 1 4 0 1 1 1 1 1 2 0 2 1 1 1 1 2 2 4 4 4
ϕ4,7 4 4 0 4 0 4 1 0 2 1 1 1 1 1 1 2 2 1 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 0 t12 0 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 0 t28 0 t20 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 t20 0 t12 0 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 t12 t20 0 1 t8 t16 0 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 t12 0 t16 1 t8 0 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 t28 0 t8 0 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 0 t15 + t21 0 t7 + t13 1 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 0 t12 + t24 0 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 t6 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 0 t9 + t27 0 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 0 t + t19 0 t11 + t17 t4 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 0 t4 + t16 0 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 0 t7 + t13 0 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 0 t5 + t23 t7 t15 + t21 0 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 0 t8 + t20 t4 t12 + t24 0 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 0 t11 + t17 t t9 + t27 0 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 0 t10 + t16 + t22 0 t8 + t14 + t26 t 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 0 t4 + t10 + t22 t6 t2 + t14 + t20 0 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 0 t8 + t14 + t26 0 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 0 t2 + t14 + t20 t4 t6 + t12 + t18 0 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 0 t6 + t18 + t24 0 t10 + t16 + t22 t3 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 0 t6 + t12 + t18 t2 t4 + t10 + t22 0 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 t7 + t13 + t19 + t25 0 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 t5 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 t5 + t11 + t17 + t23 0 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t2 0 t2 + t8 t4 t t4 t6 t3 t6 t + t7 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 − k2,1 + 2k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ2,8,  ϕ2,5,  ϕ1,20,  ϕ3,8,  ϕ3,10,  ϕ2,11,  ϕ3,6''},   {ϕ2,7,  ϕ2,4,  ϕ2,1}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,20311 + 2t
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,11322 + t
ϕ2,8942 + 4t + 3t2
ϕ2,53922 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,83933 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 4t6 + 2t7
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,101263 + 6t + 3t2
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''963 + 4t + 2t2
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1
ϕ1,8 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1
ϕ1,12 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1
ϕ1,28 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 1 1
ϕ2,15 2 2 2 2 0 2 2 0 0 0 0 0 0 0 2 1 0 0 1 1 0 2 2 2
ϕ2,12 2 2 2 2 0 2 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 2 2 2
ϕ2,9 2 2 2 2 0 2 0 0 2 0 0 1 2 0 0 0 1 2 0 0 0 2 2 2
ϕ2,11 2 2 2 2 0 2 2 0 0 1 0 0 1 0 1 0 1 0 0 2 0 2 2 2
ϕ2,8 2 2 2 2 0 2 0 1 0 0 1 0 1 0 1 1 0 1 0 0 0 2 2 2
ϕ2,5 2 2 2 2 0 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 0 2 2 2
ϕ2,7 2 2 2 2 0 2 1 0 1 0 0 0 2 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 2 2 2 2 0 2 1 0 1 0 0 0 0 1 0 0 0 1 0 1 1 2 2 2
ϕ2,1 2 2 2 2 1 2 1 0 1 0 0 1 0 0 2 2 0 0 0 0 0 2 2 2
ϕ3,2 3 3 3 3 0 3 2 0 1 0 0 1 0 1 1 2 0 0 1 1 0 3 3 3
ϕ3,8 3 3 3 3 0 3 1 0 2 0 0 1 1 1 0 0 2 2 0 1 0 3 3 3
ϕ3,4 3 3 3 3 0 3 0 1 1 0 0 2 2 0 1 1 1 2 0 0 0 3 3 3
ϕ3,10 3 3 3 3 0 3 1 1 0 0 0 0 1 0 2 1 1 0 1 2 0 3 3 3
ϕ3,6' 3 3 3 3 0 3 2 0 1 0 1 0 2 0 1 0 1 1 0 2 0 3 3 3
ϕ3,6'' 3 3 3 3 0 3 1 0 2 0 0 1 1 0 2 2 0 1 0 0 1 3 3 3
ϕ4,3 4 4 4 4 0 4 1 1 1 0 0 1 1 1 1 1 1 2 1 1 0 4 4 4
ϕ4,5 4 4 4 4 0 4 1 1 1 0 0 1 2 0 2 1 1 1 0 2 1 4 4 4
ϕ4,7 4 4 4 4 0 4 2 0 2 0 1 1 1 1 1 2 1 1 0 1 0 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 0 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 0 t3 t5 t7
ϕ1,8 t16 1 t8 t28 0 t20 0 0 t 0 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 t20 0 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 0 t2 0 t5 t7 t3
ϕ1,12 t12 t20 t28 1 0 t16 t9 0 0 0 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 t12 t20 t16 1 t8 t 0 0 0 0 0 0 0 t5 t6 0 0 0 0 0 t7 t3 t5
ϕ1,28 t20 t28 t12 t8 0 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 0 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 t + t19 t15 + t21 0 t7 + t13 1 + t6 0 0 0 0 0 0 0 t4 + t10 t5 0 0 t t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 t4 + t16 t12 + t24 0 t4 + t16 0 1 0 0 0 t5 0 t4 0 t2 t2 0 0 t6 0 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 t7 + t13 t9 + t27 0 t + t19 0 0 1 + t6 0 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 0 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 t5 + t23 t + t19 0 t11 + t17 t4 + t10 0 0 1 0 0 t2 0 t8 0 t3 0 0 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 0 t2 0 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 t11 + t17 t7 + t13 0 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 0 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 t9 + t27 t5 + t23 0 t15 + t21 t8 0 t2 0 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 t12 + t24 t5 0 t5 0 0 0 0 1 0 0 0 t6 0 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 t15 + t21 t11 + t17 t t9 + t27 t2 0 t8 0 0 t4 0 0 1 + t6 t + t7 0 0 0 0 0 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 0 t8 + t14 + t26 t + t7 0 t7 0 0 t3 0 t2 t5 1 + t6 0 0 t2 t4 0 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 0 t2 + t14 + t20 t7 0 t + t7 0 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 0 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 0 t6 + t18 + t24 0 t2 t5 0 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 0 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 0 t6 + t12 + t18 t5 t2 0 0 0 0 t3 0 t3 + t9 t4 t4 0 1 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 0 t10 + t16 + t22 t3 + t9 0 t3 0 t2 0 t + t7 0 t7 0 t2 t4 0 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 0 t4 + t10 + t22 t3 0 t3 + t9 0 0 t5 t7 0 t + t7 t2 + t8 0 t4 0 0 1 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t6 t3 t6 0 0 t2 t4 t t4 t5 t5 t + t7 t t3 0 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t4 t t4 0 0 t6 t2 + t8 0 t2 + t8 t3 t3 t5 0 t + t7 t 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 0 t t4 t6 t3 t6 t + t7 t t3 0 t5 0 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 + k2,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,8,  ϕ1,0,  ϕ2,5,  ϕ1,28,  ϕ2,11},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0111
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,282511 + 2t + 3t2 + 4t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,51442 + 3t + 4t2 + 3t3 + 2t4
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 0 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1
ϕ1,12 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1
ϕ1,28 0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 0 2 2 2 2 1 2 0 0 1 0 0 0 0 2 1 0 0 2 1 0 2 2 2
ϕ2,12 0 2 2 2 2 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 1 2 2 2
ϕ2,9 0 2 2 2 2 1 0 0 2 1 0 0 2 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 0 2 2 2 2 0 2 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 2 2
ϕ2,8 0 2 2 2 2 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 2 2 2
ϕ2,5 0 2 2 2 2 0 0 0 2 0 0 1 1 0 1 1 0 1 0 0 2 2 2 2
ϕ2,7 0 2 2 2 2 0 1 0 1 2 0 0 2 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 0 2 2 2 2 0 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 2
ϕ2,1 0 2 2 2 2 0 1 0 1 0 0 1 0 0 2 2 0 0 1 0 1 2 2 2
ϕ3,2 0 3 3 3 3 0 2 0 1 1 0 1 0 1 1 2 0 0 2 1 1 3 3 3
ϕ3,8 0 3 3 3 3 1 1 0 2 2 0 0 1 1 0 0 2 2 0 1 1 3 3 3
ϕ3,4 0 3 3 3 3 0 0 1 1 1 0 1 2 0 1 1 1 2 0 0 2 3 3 3
ϕ3,10 0 3 3 3 3 1 1 1 0 2 0 0 1 0 2 1 1 0 2 2 0 3 3 3
ϕ3,6' 0 3 3 3 3 0 2 0 1 1 1 0 2 0 1 0 2 1 1 2 0 3 3 3
ϕ3,6'' 0 3 3 3 3 1 1 0 2 0 1 0 1 0 2 2 0 1 1 0 2 3 3 3
ϕ4,3 0 4 4 4 4 0 1 1 1 2 0 1 1 1 1 1 1 2 2 1 1 4 4 4
ϕ4,5 0 4 4 4 4 1 1 1 1 1 1 0 2 0 2 1 1 1 1 2 2 4 4 4
ϕ4,7 0 4 4 4 4 1 2 0 2 1 1 0 1 1 1 2 2 1 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 t20 0 0 0 t9 0 0 0 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 0 1 t8 t28 t12 0 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 0 t16 1 t20 t28 0 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 0 t20 t28 1 t8 0 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 0 t12 t20 t16 1 t8 t 0 0 0 0 0 0 0 t5 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 0 t28 t12 t8 t16 1 0 0 t5 0 0 0 t9 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 0 t11 + t17 t + t19 t15 + t21 t5 + t23 t7 1 + t6 0 0 t2 0 0 0 0 t4 + t10 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 0 t8 + t20 t4 + t16 t12 + t24 t8 + t20 t4 0 1 0 t5 0 0 0 t4 0 t2 t2 0 0 t6 t6 t3 + t9 2t5 t + t7
ϕ2,9 0 t5 + t23 t7 + t13 t9 + t27 t11 + t17 t 0 0 1 + t6 t8 0 0 t4 + t10 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 0 t15 + t21 t5 + t23 t + t19 t9 + t27 0 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 0 t12 + t24 t8 + t20 t4 + t16 t12 + t24 0 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 0 t9 + t27 t11 + t17 t7 + t13 t15 + t21 0 0 0 t4 + t10 0 0 1 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 0 t + t19 t9 + t27 t5 + t23 t7 + t13 0 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 0 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 0 t7 + t13 t15 + t21 t11 + t17 t + t19 0 t2 0 t8 0 0 t4 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 0 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 0 t + t7 0 t7 t3 0 t3 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 0 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 t2 t7 0 t + t7 t3 + t9 0 0 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 0 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 0 0 t2 t5 t7 0 t t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 0 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 t6 t5 t2 0 t + t7 0 0 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' 0 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 0 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' 0 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 t4 t3 0 t3 + t9 0 t2 0 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t6 t3 t6 t2 + t8 0 t2 t4 t t4 t5 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 t4 t t4 t6 t3 0 t2 + t8 0 t2 + t8 t3 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 t2 + t8 0 t2 + t8 t4 t 0 t6 t3 t6 t + t7 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 + k2,1 − 2k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,8,  ϕ2,5,  ϕ1,8,  ϕ3,2,  ϕ3,4,  ϕ3,6',  ϕ2,11},   {ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,8311 + 2t
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,113922 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ2,8942 + 4t + 3t2
ϕ2,5322 + t
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,23933 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 4t6 + 2t7
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,41263 + 6t + 3t2
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'963 + 4t + 2t2
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1
ϕ1,16 1 0 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 1 1
ϕ1,12 1 0 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1
ϕ1,20 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1
ϕ1,28 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1
ϕ2,15 2 0 2 2 2 2 2 0 0 1 0 0 0 0 2 1 0 0 2 0 0 2 2 2
ϕ2,12 2 0 2 2 2 2 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 2 2 2
ϕ2,9 2 0 2 2 2 2 0 0 2 0 0 0 2 0 0 0 1 1 0 0 1 2 2 2
ϕ2,11 2 0 2 2 2 2 2 0 0 2 0 0 1 0 1 0 1 0 1 0 0 2 2 2
ϕ2,8 2 0 2 2 2 2 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 2 2 2
ϕ2,5 2 0 2 2 2 2 0 0 2 0 0 1 1 0 1 1 0 0 0 0 2 2 2 2
ϕ2,7 2 1 2 2 2 2 1 0 1 1 0 0 2 0 0 0 2 0 0 0 0 2 2 2
ϕ2,4 2 0 2 2 2 2 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 2 2 2
ϕ2,1 2 0 2 2 2 2 1 0 1 0 0 0 0 0 2 2 0 0 1 0 1 2 2 2
ϕ3,2 3 0 3 3 3 3 2 0 1 1 0 0 0 1 1 2 0 0 2 0 1 3 3 3
ϕ3,8 3 0 3 3 3 3 1 0 2 1 0 0 1 1 0 0 2 1 0 0 1 3 3 3
ϕ3,4 3 0 3 3 3 3 0 1 1 0 0 0 2 0 1 1 1 1 0 0 2 3 3 3
ϕ3,10 3 0 3 3 3 3 1 1 0 2 0 0 1 0 2 1 1 0 2 0 0 3 3 3
ϕ3,6' 3 0 3 3 3 3 2 0 1 1 0 0 2 0 1 0 2 0 1 1 0 3 3 3
ϕ3,6'' 3 0 3 3 3 3 1 0 2 0 1 0 1 0 2 1 0 0 1 0 2 3 3 3
ϕ4,3 4 0 4 4 4 4 1 1 1 1 0 0 1 1 1 1 1 1 2 0 1 4 4 4
ϕ4,5 4 0 4 4 4 4 1 1 1 1 0 0 2 0 2 1 1 0 1 1 2 4 4 4
ϕ4,7 4 0 4 4 4 4 2 0 2 1 1 0 1 1 1 1 2 0 1 0 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 t16 t12 t20 t28 0 0 t9 0 0 0 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 t8 t28 t12 t20 0 0 t 0 0 0 t5 0 0 0 t6 0 0 0 0 t7 t3 t5
ϕ1,16 t8 0 1 t20 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 0 0 t5 t7 t3
ϕ1,12 t12 0 t28 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 0 0 t3 t5 t7
ϕ1,20 t28 0 t20 t16 1 t8 t 0 0 0 0 0 0 0 t5 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 0 t12 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 0 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 0 t + t19 t15 + t21 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 0 0 0 t4 + t10 t5 0 0 t + t7 0 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 0 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 1 0 t5 0 0 0 t4 0 t2 t2 0 0 0 t6 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 0 t7 + t13 t9 + t27 t11 + t17 t + t19 0 0 1 + t6 0 0 0 t4 + t10 0 0 0 t5 t 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 0 t5 + t23 t + t19 t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 0 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 0 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 t4 0 0 1 0 t5 0 t5 0 t6 0 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 0 t11 + t17 t7 + t13 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 t8 0 t2 t3 0 0 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t t9 + t27 t5 + t23 t7 + t13 t15 + t21 t8 0 t2 t4 0 0 1 + t6 0 0 0 t + t7 0 0 0 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 0 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t5 0 t5 0 0 0 0 1 0 0 0 0 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 0 t15 + t21 t11 + t17 t + t19 t9 + t27 t2 0 t8 0 0 0 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 0 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 0 0 t2 t5 1 + t6 0 0 t2 + t8 0 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 0 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 0 0 t5 t2 0 0 1 + t6 t2 0 0 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 0 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 0 0 0 t3 + t9 0 t3 t4 t4 1 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 0 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 0 t3 0 t3 + t9 t4 t4 0 1 + t6 0 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 0 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 0 0 t + t7 0 t7 0 t2 + t8 0 t4 1 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 0 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 0 t7 0 t + t7 t2 0 0 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 0 0 t4 t t4 t5 t5 t t + t7 0 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 0 0 t2 + t8 0 t2 + t8 t3 t3 0 t5 t t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t 0 t6 t3 t6 t t + t7 0 t3 0 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 + k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ1,8,  ϕ1,28,  ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,82511 + 2t + 3t2 + 4t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,28111
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,91442 + 3t + 4t2 + 3t3 + 2t4
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1
ϕ1,8 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1
ϕ1,16 1 0 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1
ϕ1,12 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 1
ϕ1,28 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1
ϕ2,15 2 0 2 2 2 0 2 0 0 1 0 1 0 0 2 1 0 0 2 1 0 2 2 2
ϕ2,12 2 0 2 2 2 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 2 2 2
ϕ2,9 2 0 2 2 2 0 0 0 1 1 0 1 2 0 0 0 1 2 0 0 1 2 2 2
ϕ2,11 2 0 2 2 2 0 2 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 2 2
ϕ2,8 2 0 2 2 2 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 2 2 2
ϕ2,5 2 0 2 2 2 0 0 0 1 0 0 2 1 0 1 1 0 1 0 0 2 2 2 2
ϕ2,7 2 1 2 2 2 0 1 0 0 2 0 0 2 0 0 0 2 1 0 1 0 2 2 2
ϕ2,4 2 1 2 2 2 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 2 2 2
ϕ2,1 2 1 2 2 2 0 1 0 0 0 0 2 0 0 2 2 0 0 1 0 1 2 2 2
ϕ3,2 3 1 3 3 3 0 2 0 0 1 0 2 0 1 1 2 0 0 2 1 1 3 3 3
ϕ3,8 3 0 3 3 3 0 1 0 1 2 0 1 1 1 0 0 2 2 0 1 1 3 3 3
ϕ3,4 3 1 3 3 3 0 0 1 0 1 0 2 2 0 1 1 1 2 0 0 2 3 3 3
ϕ3,10 3 0 3 3 3 0 1 1 0 2 0 1 1 0 2 1 1 0 2 2 0 3 3 3
ϕ3,6' 3 1 3 3 3 0 2 0 0 1 1 0 2 0 1 0 2 1 1 2 0 3 3 3
ϕ3,6'' 3 0 3 3 3 0 1 0 1 0 1 1 1 0 2 2 0 1 1 0 2 3 3 3
ϕ4,3 4 1 4 4 4 0 1 1 0 2 0 2 1 1 1 1 1 2 2 1 1 4 4 4
ϕ4,5 4 1 4 4 4 0 1 1 0 1 1 1 2 0 2 1 1 1 1 2 2 4 4 4
ϕ4,7 4 0 4 4 4 0 2 0 1 1 1 1 1 1 1 2 2 1 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 t20 0 0 0 0 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 t7
ϕ1,8 t16 1 t8 t28 t12 0 0 0 0 t9 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 0 1 t20 t28 0 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 t12 0 t28 1 t8 0 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 0 t20 t16 1 0 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 0 t12 t8 t16 1 0 0 0 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 0 t + t19 t15 + t21 t5 + t23 0 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 0 t4 + t16 t12 + t24 t8 + t20 0 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 t6 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 0 t7 + t13 t9 + t27 t11 + t17 0 0 0 1 t8 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 0 t5 + t23 t + t19 t9 + t27 0 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 0 t8 + t20 t4 + t16 t12 + t24 0 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 0 t11 + t17 t7 + t13 t15 + t21 0 0 0 t4 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t t9 + t27 t5 + t23 t7 + t13 0 t8 0 0 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 t12 + t24 t8 + t20 t4 + t16 0 t5 0 0 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 t15 + t21 t11 + t17 t + t19 0 t2 0 0 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 0 t + t7 0 0 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 0 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 0 t7 0 t t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 0 0 t2 0 t7 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 0 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 0 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 0 t3 + t9 0 0 t5 t2 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 0 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 0 t3 0 t3 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t6 t3 0 t2 + t8 0 t2 + t8 t4 t t4 t5 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t4 t 0 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t2 + t8 0 t2 t4 t t4 t6 t3 t6 t + t7 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 + k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ1,0,  ϕ3,2,  ϕ2,15,  ϕ3,4,  ϕ2,12,  ϕ2,9,  ϕ3,6'},   {ϕ2,8,  ϕ2,5,  ϕ2,11}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0311 + 2t
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,15322 + t
ϕ2,12942 + 4t + 3t2
ϕ2,93922 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,21263 + 6t + 3t2
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,4963 + 4t + 2t2
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'3933 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 4t6 + 2t7
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 1
ϕ1,8 0 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1
ϕ1,16 0 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1
ϕ1,12 0 1 1 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1
ϕ1,20 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1
ϕ1,28 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1
ϕ2,15 0 2 2 2 2 2 1 0 0 1 0 1 0 0 2 0 0 0 2 1 0 2 2 2
ϕ2,12 0 2 2 2 2 2 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 2 2 2
ϕ2,9 0 2 2 2 2 2 0 0 2 1 0 1 2 0 0 0 1 0 0 0 1 2 2 2
ϕ2,11 0 2 2 2 2 2 0 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 2 2
ϕ2,8 0 2 2 2 2 2 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 2 2 2
ϕ2,5 1 2 2 2 2 2 0 0 1 0 0 2 1 0 1 0 0 0 0 0 2 2 2 2
ϕ2,7 0 2 2 2 2 2 0 0 1 2 0 0 2 0 0 0 2 0 0 1 0 2 2 2
ϕ2,4 0 2 2 2 2 2 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 2 2 2
ϕ2,1 0 2 2 2 2 2 0 0 0 0 0 2 0 0 2 1 0 0 1 0 1 2 2 2
ϕ3,2 0 3 3 3 3 3 0 0 0 1 0 2 0 1 1 1 0 0 2 1 1 3 3 3
ϕ3,8 0 3 3 3 3 3 0 0 2 2 0 1 1 1 0 0 2 0 0 1 1 3 3 3
ϕ3,4 0 3 3 3 3 3 0 0 1 1 0 2 2 0 1 0 1 1 0 0 2 3 3 3
ϕ3,10 0 3 3 3 3 3 0 1 0 2 0 1 1 0 2 0 1 0 2 1 0 3 3 3
ϕ3,6' 0 3 3 3 3 3 0 0 1 1 1 0 2 0 1 0 2 0 1 2 0 3 3 3
ϕ3,6'' 0 3 3 3 3 3 0 0 1 0 1 1 1 0 2 1 0 0 1 0 2 3 3 3
ϕ4,3 0 4 4 4 4 4 0 0 1 2 0 2 1 1 1 0 1 1 2 1 1 4 4 4
ϕ4,5 0 4 4 4 4 4 0 1 1 1 1 1 2 0 2 0 1 0 1 1 2 4 4 4
ϕ4,7 0 4 4 4 4 4 0 0 1 1 1 1 1 1 1 1 2 0 1 1 1 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 t20 t28 0 0 0 0 0 t5 0 0 t 0 0 0 0 0 t6 t3 t5 t7
ϕ1,8 0 1 t8 t28 t12 t20 0 0 t t9 0 0 t5 0 0 0 t6 0 0 0 0 t7 t3 t5
ϕ1,16 0 t16 1 t20 t28 t12 0 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 0 t20 t28 1 t8 t16 0 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 t7
ϕ1,20 0 t12 t20 t16 1 t8 t 0 0 0 0 t9 0 0 t5 0 0 0 t2 0 0 t7 t3 t5
ϕ1,28 0 t28 t12 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 0 0 0 t2 t5 t7 t3
ϕ2,15 0 t11 + t17 t + t19 t15 + t21 t5 + t23 t7 + t13 1 0 0 t2 0 t8 0 0 t4 + t10 0 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 0 t8 + t20 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 0 t2 0 0 0 t6 t3 + t9 2t5 t + t7
ϕ2,9 0 t5 + t23 t7 + t13 t9 + t27 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 0 0 0 t3 2t6 t2 + t8 2t4
ϕ2,11 0 t15 + t21 t5 + t23 t + t19 t9 + t27 t11 + t17 0 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 0 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 0 0 0 1 0 t5 0 t5 0 t6 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t t9 + t27 t11 + t17 t7 + t13 t15 + t21 t5 + t23 0 0 t4 0 0 1 + t6 t8 0 t2 0 0 0 0 0 t + t7 2t4 2t6 t2 + t8
ϕ2,7 0 t + t19 t9 + t27 t5 + t23 t7 + t13 t15 + t21 0 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 0 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 0 t4 + t16 t12 + t24 t8 + t20 t4 + t16 t12 + t24 0 0 t5 0 t4 0 0 1 0 0 0 0 t6 t2 t2 2t5 t + t7 t3 + t9
ϕ2,1 0 t7 + t13 t15 + t21 t11 + t17 t + t19 t9 + t27 0 0 0 0 0 t4 + t10 0 0 1 + t6 t 0 0 t3 0 t5 t2 + t8 2t4 2t6
ϕ3,2 0 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 0 0 0 t3 0 t3 + t9 0 t2 t5 1 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,8 0 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 0 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 0 0 t4 t4 t + 2t7 2t3 + t9 3t5
ϕ3,4 0 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 0 t5 t7 0 t + t7 t3 + t9 0 t3 0 t4 1 0 0 t2 + t8 3t5 t + 2t7 2t3 + t9
ϕ3,10 0 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 0 t2 0 t + t7 0 t7 t3 0 t3 + t9 0 t4 0 1 + t6 t2 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' 0 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 0 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 0 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' 0 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 0 0 t3 0 t2 t5 t7 0 t + t7 t2 0 0 t4 0 1 + t6 2t3 + t9 3t5 t + 2t7
ϕ4,3 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 0 t6 t2 + t8 0 t2 + t8 t4 t t4 0 t5 t t + t7 t3 t3 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 0 t3 0 t5 t t + t7 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 0 t2 t4 t t4 t6 t3 t6 t t + t7 0 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane k1,1 + 2k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,7,  ϕ2,4,  ϕ2,1,  ϕ1,28,  ϕ3,8,  ϕ3,10,  ϕ3,6''}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,28311 + 2t
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,73922 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ2,4942 + 4t + 3t2
ϕ2,1322 + t
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,8963 + 4t + 2t2
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,103933 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 4t6 + 2t7
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''1263 + 6t + 3t2
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1
ϕ1,8 1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 1
ϕ1,16 1 1 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1
ϕ1,12 1 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1
ϕ1,20 1 1 1 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1
ϕ1,28 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 1
ϕ2,15 2 2 2 2 2 0 2 0 0 1 0 1 0 0 0 1 0 0 2 1 0 2 2 2
ϕ2,12 2 2 2 2 2 0 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 2 2 2
ϕ2,9 2 2 2 2 2 1 0 0 2 1 0 1 1 0 0 0 0 2 0 0 0 2 2 2
ϕ2,11 2 2 2 2 2 0 2 0 0 2 0 0 1 0 0 0 0 0 1 2 0 2 2 2
ϕ2,8 2 2 2 2 2 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 2 2 2
ϕ2,5 2 2 2 2 2 0 0 0 2 0 0 2 0 0 0 1 0 1 0 0 1 2 2 2
ϕ2,7 2 2 2 2 2 0 1 0 1 2 0 0 2 0 0 0 0 1 0 1 0 2 2 2
ϕ2,4 2 2 2 2 2 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 2 2 2
ϕ2,1 2 2 2 2 2 0 1 0 1 0 0 2 0 0 1 2 0 0 1 0 0 2 2 2
ϕ3,2 3 3 3 3 3 0 2 0 1 1 0 2 0 1 0 2 0 0 1 1 0 3 3 3
ϕ3,8 3 3 3 3 3 0 1 0 2 2 0 1 1 0 0 0 1 2 0 1 0 3 3 3
ϕ3,4 3 3 3 3 3 0 0 1 1 1 0 2 1 0 0 1 0 2 0 0 1 3 3 3
ϕ3,10 3 3 3 3 3 0 1 1 0 2 0 1 1 0 0 1 0 0 2 2 0 3 3 3
ϕ3,6' 3 3 3 3 3 0 2 0 1 1 1 0 2 0 0 0 0 1 1 2 0 3 3 3
ϕ3,6'' 3 3 3 3 3 0 1 0 2 0 1 1 0 0 0 2 0 1 1 0 1 3 3 3
ϕ4,3 4 4 4 4 4 0 1 1 1 2 0 2 1 1 0 1 0 2 1 1 0 4 4 4
ϕ4,5 4 4 4 4 4 0 1 1 1 1 1 1 1 0 0 1 0 1 1 2 1 4 4 4
ϕ4,7 4 4 4 4 4 0 2 0 2 1 1 1 1 0 0 2 1 1 1 1 0 4 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 t20 0 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 0 t3 t5 t7
ϕ1,8 t16 1 t8 t28 t12 0 0 0 t t9 0 0 t5 0 0 0 0 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 t20 t28 0 t5 0 0 t 0 0 0 0 0 0 0 0 t6 t2 0 t5 t7 t3
ϕ1,12 t12 t20 t28 1 t8 0 t9 0 0 t5 0 0 t 0 0 0 0 0 0 t6 0 t3 t5 t7
ϕ1,20 t28 t12 t20 t16 1 0 t 0 0 0 0 t9 0 0 0 t6 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 t28 t12 t8 t16 1 0 0 t5 0 0 t 0 0 0 0 0 t6 0 0 0 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 t + t19 t15 + t21 t5 + t23 0 1 + t6 0 0 t2 0 t8 0 0 0 t5 0 0 t + t7 t3 0 2t6 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 0 1 0 t5 0 t5 0 0 0 t2 t2 0 0 t6 0 t3 + t9 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 t7 + t13 t9 + t27 t11 + t17 t 0 0 1 + t6 t8 0 t2 t4 0 0 0 0 t + t7 0 0 0 2t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 t5 + t23 t + t19 t9 + t27 0 t4 + t10 0 0 1 + t6 0 0 t2 0 0 0 0 0 t5 t + t7 0 2t4 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 t8 + t20 t4 + t16 t12 + t24 0 0 t4 0 0 1 0 t5 0 0 t6 0 t2 t2 0 0 t + t7 t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 t11 + t17 t7 + t13 t15 + t21 0 0 0 t4 + t10 0 0 1 + t6 0 0 0 t3 0 t5 0 0 t 2t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 t9 + t27 t5 + t23 t7 + t13 0 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 0 t3 0 t5 0 t2 + t8 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 t5 0 t5 0 t4 0 0 1 0 0 0 t6 0 t2 0 2t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 t15 + t21 t11 + t17 t + t19 0 t2 0 t8 0 0 t4 + t10 0 0 1 t + t7 0 0 t3 0 0 t2 + t8 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 0 t + t7 0 t7 t3 0 t3 + t9 0 t2 0 1 + t6 0 0 t2 t4 0 t + 2t7 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 0 t7 0 t + t7 t3 + t9 0 t3 t5 0 0 0 1 t2 + t8 0 t4 0 t + 2t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 0 0 t2 t5 t7 0 t + t7 t3 0 0 t4 0 1 + t6 0 0 t2 3t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 0 t5 t2 0 t + t7 0 t7 t3 0 0 t4 0 0 1 + t6 t2 + t8 0 3t5 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 0 t3 + t9 0 t3 t5 t2 0 t + t7 0 0 0 0 t4 t4 1 + t6 0 2t3 + t9 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 0 t3 0 t3 + t9 0 t2 t5 0 0 0 t2 + t8 0 t4 t4 0 1 2t3 + t9 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t 0 t5 0 t + t7 t t3 0 1 + 3t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t4 t t4 t6 t3 t6 t2 0 0 t3 0 t5 t5 t + t7 t 3t4 + t10 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t2 + t8 0 t2 + t8 t4 t t4 t6 0 0 t + t7 t t3 t3 t5 0 2t2 + 2t8 3t4 + t10 1 + 3t6

For the generic point of the hyperplane 2k1,1 − 2k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ4,7,  ϕ1,16,  ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,16611 + 2t + 3t2
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,8633 + 2t + t2
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104233 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 5t6 + 4t7
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,74244 + 5t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 0 1 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0
ϕ1,8 1 1 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 1
ϕ1,16 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0
ϕ1,12 1 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0
ϕ1,20 1 1 0 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0
ϕ1,28 1 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0
ϕ2,15 2 2 1 2 2 2 2 0 0 1 0 1 0 0 2 1 0 0 1 1 0 2 2 0
ϕ2,12 2 2 0 2 2 2 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 2 2 1
ϕ2,9 2 2 0 2 2 2 0 0 2 1 0 1 2 0 0 0 0 2 0 0 1 2 2 1
ϕ2,11 2 2 0 2 2 2 2 0 0 2 0 0 1 0 1 0 0 0 1 2 0 2 2 1
ϕ2,8 2 2 0 2 2 2 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 2 2 1
ϕ2,5 2 2 0 2 2 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 2 2 2 0
ϕ2,7 2 2 0 2 2 2 1 0 1 2 0 0 2 0 0 0 1 1 0 1 0 2 2 1
ϕ2,4 2 2 0 2 2 2 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 0
ϕ2,1 2 2 0 2 2 2 1 0 1 0 0 2 0 0 2 2 0 0 1 0 1 2 2 0
ϕ3,2 3 3 1 3 3 3 2 0 1 1 0 2 0 1 1 2 0 0 1 1 1 3 3 0
ϕ3,8 3 3 0 3 3 3 1 0 2 2 0 1 1 1 0 0 1 2 0 1 1 3 3 1
ϕ3,4 3 3 0 3 3 3 0 1 1 1 0 2 2 0 1 1 0 2 0 0 2 3 3 1
ϕ3,10 3 3 0 3 3 3 1 1 0 2 0 1 1 0 2 1 0 0 2 2 0 3 3 1
ϕ3,6' 3 3 0 3 3 3 2 0 1 1 1 0 2 0 1 0 0 1 1 2 0 3 3 2
ϕ3,6'' 3 3 0 3 3 3 1 0 2 0 1 1 1 0 2 2 0 1 1 0 2 3 3 0
ϕ4,3 4 4 0 4 4 4 1 1 1 2 0 2 1 1 1 1 0 2 2 1 1 4 4 1
ϕ4,5 4 4 0 4 4 4 1 1 1 1 1 1 2 0 2 1 0 1 1 2 2 4 4 1
ϕ4,7 4 4 0 4 4 4 2 0 2 1 1 1 1 1 1 2 0 1 1 1 1 4 4 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 0 t12 t20 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 0
ϕ1,8 t16 1 0 t28 t12 t20 0 0 t t9 0 0 t5 0 0 0 0 t2 0 0 0 t7 t3 t5
ϕ1,16 t8 t16 1 t20 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 0 t2 0 t5 t7 0
ϕ1,12 t12 t20 0 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 0
ϕ1,20 t28 t12 0 t16 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 t7 t3 0
ϕ1,28 t20 t28 0 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 t7 0
ϕ2,15 t9 + t27 t11 + t17 t t15 + t21 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t t3 0 2t6 t2 + t8 0
ϕ2,12 t12 + t24 t8 + t20 0 t12 + t24 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 0 0 0 t6 t6 t3 + t9 2t5 t
ϕ2,9 t15 + t21 t5 + t23 0 t9 + t27 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 0 t + t7 0 0 t3 2t6 t2 + t8 t4
ϕ2,11 t7 + t13 t15 + t21 0 t + t19 t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 0 0 t5 t + t7 0 2t4 2t6 t2
ϕ2,8 t4 + t16 t12 + t24 0 t4 + t16 t12 + t24 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 0 t2 t2 0 0 t + t7 t3 + t9 t5
ϕ2,5 t + t19 t9 + t27 0 t7 + t13 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 2t6 0
ϕ2,7 t11 + t17 t + t19 0 t5 + t23 t7 + t13 t15 + t21 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t t3 0 t5 0 t2 + t8 2t4 t6
ϕ2,4 t8 + t20 t4 + t16 0 t8 + t20 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 0
ϕ2,1 t5 + t23 t7 + t13 0 t11 + t17 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 t5 t2 + t8 2t4 0
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 t4 t4 t + 2t7 2t3 + t9 0
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 0 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 t2 + t8 0 t4 t4 t + 2t7 2t3 + t9 t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 0 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 0 1 + t6 0 0 t2 + t8 3t5 t + 2t7 t3
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 0 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 0 0 1 + t6 t2 + t8 0 3t5 t + 2t7 t3
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 0 t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 0 t4 t4 1 + t6 0 2t3 + t9 3t5 t + t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 0 t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 + t6 2t3 + t9 3t5 0
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 t5 0 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 t4
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 0 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 t2
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 t + t7 0 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + t6

For the generic point of the hyperplane 2k1,1 − k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ4,3,  ϕ1,12,  ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,12611 + 2t + 3t2
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,4633 + 2t + t2
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4233 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 5t6 + 4t7
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,34244 + 5t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1
ϕ1,8 1 1 1 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 1
ϕ1,16 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1
ϕ1,12 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1
ϕ1,20 1 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1
ϕ1,28 1 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1
ϕ2,15 2 2 2 0 2 2 2 0 0 1 0 1 0 0 2 1 0 0 2 1 0 0 2 2
ϕ2,12 2 2 2 0 2 2 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 2 2
ϕ2,9 2 2 2 0 2 2 0 0 2 1 0 1 2 0 0 0 1 1 0 0 1 1 2 2
ϕ2,11 2 2 2 1 2 2 2 0 0 2 0 0 1 0 1 0 1 0 1 1 0 0 2 2
ϕ2,8 2 2 2 0 2 2 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 2 2
ϕ2,5 2 2 2 0 2 2 0 0 2 0 0 2 1 0 1 1 0 0 0 0 2 1 2 2
ϕ2,7 2 2 2 0 2 2 1 0 1 2 0 0 2 0 0 0 2 0 0 1 0 1 2 2
ϕ2,4 2 2 2 0 2 2 1 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 2 2
ϕ2,1 2 2 2 0 2 2 1 0 1 0 0 2 0 0 2 2 0 0 1 0 1 0 2 2
ϕ3,2 3 3 3 0 3 3 2 0 1 1 0 2 0 1 1 2 0 0 2 1 1 0 3 3
ϕ3,8 3 3 3 0 3 3 1 0 2 2 0 1 1 1 0 0 2 0 0 1 1 2 3 3
ϕ3,4 3 3 3 0 3 3 0 1 1 1 0 2 2 0 1 1 1 1 0 0 2 1 3 3
ϕ3,10 3 3 3 1 3 3 1 1 0 2 0 1 1 0 2 1 1 0 2 1 0 0 3 3
ϕ3,6' 3 3 3 0 3 3 2 0 1 1 1 0 2 0 1 0 2 0 1 2 0 1 3 3
ϕ3,6'' 3 3 3 0 3 3 1 0 2 0 1 1 1 0 2 2 0 0 1 0 2 1 3 3
ϕ4,3 4 4 4 0 4 4 1 1 1 2 0 2 1 1 1 1 1 0 2 1 1 2 4 4
ϕ4,5 4 4 4 0 4 4 1 1 1 1 1 1 2 0 2 1 1 0 1 2 2 1 4 4
ϕ4,7 4 4 4 0 4 4 2 0 2 1 1 1 1 1 1 2 2 0 1 1 1 1 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 0 t20 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 0 t5 t7
ϕ1,8 t16 1 t8 0 t12 t20 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 0 t3 t5
ϕ1,16 t8 t16 1 0 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 0 t7 t3
ϕ1,12 t12 t20 t28 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 0 0 0 t5 t7
ϕ1,20 t28 t12 t20 0 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 0 t3 t5
ϕ1,28 t20 t28 t12 0 t16 1 0 0 t5 0 0 t t9 0 0 0 0 0 0 0 t2 t5 t7 t3
ϕ2,15 t9 + t27 t11 + t17 t + t19 0 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t + t7 t3 0 0 t2 + t8 2t4
ϕ2,12 t12 + t24 t8 + t20 t4 + t16 0 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 t6 0 2t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 t7 + t13 0 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t 0 0 t3 t6 t2 + t8 2t4
ϕ2,11 t7 + t13 t15 + t21 t5 + t23 t t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t 0 0 2t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 t8 + t20 0 t12 + t24 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 t6 0 t2 0 0 t t3 + t9 2t5
ϕ2,5 t + t19 t9 + t27 t11 + t17 0 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 0 0 0 t + t7 t4 2t6 t2 + t8
ϕ2,7 t11 + t17 t + t19 t9 + t27 0 t7 + t13 t15 + t21 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 0 0 t5 0 t2 2t4 2t6
ϕ2,4 t8 + t20 t4 + t16 t12 + t24 0 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 0 t6 t2 t2 t5 t + t7 t3 + t9
ϕ2,1 t5 + t23 t7 + t13 t15 + t21 0 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 t5 0 2t4 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 0 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 t4 t4 0 2t3 + t9 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 0 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 0 0 t4 t4 t + t7 2t3 + t9 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 0 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 t4 1 0 0 t2 + t8 t5 t + 2t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 t2 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 0 0 t + 2t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 0 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 0 t4 1 + t6 0 t3 3t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 0 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 + t8 0 0 t4 0 1 + t6 t3 3t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 t5 t5 0 t + t7 t3 t3 1 + t6 2t2 + 2t8 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 0 t5 t + t7 t + t7 t4 1 + 3t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 t + t7 t + t7 0 t3 t5 t5 t2 3t4 + t10 1 + 3t6

For the generic point of the hyperplane 2k1,1 − k2,1 + 2k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ4,5,  ϕ1,20,  ϕ3,2,  ϕ3,4,  ϕ3,6'}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,20611 + 2t + 3t2
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24233 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 5t6 + 4t7
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'633 + 2t + t2
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,54244 + 5t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1
ϕ1,8 1 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1
ϕ1,16 1 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1
ϕ1,12 1 1 1 1 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1
ϕ1,20 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1
ϕ1,28 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1
ϕ2,15 2 2 2 2 0 2 2 0 0 1 0 1 0 0 2 1 0 0 2 0 0 2 1 2
ϕ2,12 2 2 2 2 0 2 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 2 1 2
ϕ2,9 2 2 2 2 0 2 0 0 2 1 0 1 2 0 0 0 1 2 0 0 1 2 0 2
ϕ2,11 2 2 2 2 0 2 2 0 0 2 0 0 1 0 1 0 1 0 1 1 0 2 1 2
ϕ2,8 2 2 2 2 0 2 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 2 0 2
ϕ2,5 2 2 2 2 0 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 2 2 0 2
ϕ2,7 2 2 2 2 0 2 1 0 1 2 0 0 2 0 0 0 2 1 0 0 0 2 1 2
ϕ2,4 2 2 2 2 0 2 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 2 1 2
ϕ2,1 2 2 2 2 1 2 1 0 1 0 0 2 0 0 2 1 0 0 1 0 1 2 0 2
ϕ3,2 3 3 3 3 0 3 2 0 1 1 0 2 0 1 1 2 0 0 2 0 1 3 1 3
ϕ3,8 3 3 3 3 0 3 1 0 2 2 0 1 1 1 0 0 2 2 0 0 1 3 1 3
ϕ3,4 3 3 3 3 0 3 0 1 1 1 0 2 2 0 1 1 1 2 0 0 2 3 0 3
ϕ3,10 3 3 3 3 0 3 1 1 0 2 0 1 1 0 2 1 1 0 2 0 0 3 2 3
ϕ3,6' 3 3 3 3 0 3 2 0 1 1 1 0 2 0 1 0 2 1 1 1 0 3 1 3
ϕ3,6'' 3 3 3 3 1 3 1 0 2 0 1 1 1 0 2 1 0 1 1 0 2 3 0 3
ϕ4,3 4 4 4 4 0 4 1 1 1 2 0 2 1 1 1 1 1 2 2 0 1 4 1 4
ϕ4,5 4 4 4 4 0 4 1 1 1 1 1 1 2 0 2 1 1 1 1 0 2 4 2 4
ϕ4,7 4 4 4 4 0 4 2 0 2 1 1 1 1 1 1 2 2 1 1 0 1 4 1 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 0 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 0 t7
ϕ1,8 t16 1 t8 t28 0 t20 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 t7 0 t5
ϕ1,16 t8 t16 1 t20 0 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 0 t3
ϕ1,12 t12 t20 t28 1 0 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 0 0 t3 t5 t7
ϕ1,20 t28 t12 t20 t16 1 t8 t 0 0 0 0 t9 0 0 t5 0 0 0 t2 0 0 t7 0 t5
ϕ1,28 t20 t28 t12 t8 0 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 0 t3
ϕ2,15 t9 + t27 t11 + t17 t + t19 t15 + t21 0 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t + t7 0 0 2t6 t2 2t4
ϕ2,12 t12 + t24 t8 + t20 t4 + t16 t12 + t24 0 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 0 t6 t3 + t9 t5 t + t7
ϕ2,9 t15 + t21 t5 + t23 t7 + t13 t9 + t27 0 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 2t6 0 2t4
ϕ2,11 t7 + t13 t15 + t21 t5 + t23 t + t19 0 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t 0 2t4 t6 t2 + t8
ϕ2,8 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 0 2t5
ϕ2,5 t + t19 t9 + t27 t11 + t17 t7 + t13 0 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t + t7 2t4 0 t2 + t8
ϕ2,7 t11 + t17 t + t19 t9 + t27 t5 + t23 0 t15 + t21 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 0 0 t2 + t8 t4 2t6
ϕ2,4 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 0 t2 2t5 t t3 + t9
ϕ2,1 t5 + t23 t7 + t13 t15 + t21 t11 + t17 t t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t 0 0 t3 0 t5 t2 + t8 0 2t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 0 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 0 t4 t + 2t7 t3 3t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 0 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 0 t4 t + 2t7 t3 3t5
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 0 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 + t8 3t5 0 2t3 + t9
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 0 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 t4 0 1 + t6 0 0 3t5 t + t7 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 0 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 0 2t3 + t9 t5 t + 2t7
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 t2 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 0 t4 t4 0 1 + t6 2t3 + t9 0 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 t5 t5 t + t7 t + t7 0 t3 1 + 3t6 t2 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 t5 t5 0 t + t7 3t4 + t10 1 + t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 t + t7 t + t7 t3 t3 0 t5 2t2 + 2t8 t4 1 + 3t6

For the generic point of the hyperplane 2k1,1 + k2,1 − 2k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ4,5,  ϕ1,8,  ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,7,  ϕ2,4,  ϕ2,1}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,8611 + 2t + 3t2
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84233 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 5t6 + 4t7
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''633 + 2t + t2
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,54244 + 5t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 1 1 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1
ϕ1,8 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1
ϕ1,16 1 0 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1
ϕ1,12 1 0 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1
ϕ1,20 1 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1
ϕ1,28 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1
ϕ2,15 2 0 2 2 2 2 2 0 0 1 0 1 0 0 2 1 0 0 2 1 0 2 0 2
ϕ2,12 2 0 2 2 2 2 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 2 1 2
ϕ2,9 2 0 2 2 2 2 0 0 2 1 0 1 2 0 0 0 1 2 0 0 0 2 1 2
ϕ2,11 2 0 2 2 2 2 2 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 0 2
ϕ2,8 2 0 2 2 2 2 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 2 0 2
ϕ2,5 2 0 2 2 2 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 1 2 1 2
ϕ2,7 2 1 2 2 2 2 1 0 1 2 0 0 2 0 0 0 1 1 0 1 0 2 0 2
ϕ2,4 2 0 2 2 2 2 1 0 1 0 1 0 0 1 0 0 0 1 1 1 0 2 1 2
ϕ2,1 2 0 2 2 2 2 1 0 1 0 0 2 0 0 2 2 0 0 1 0 0 2 1 2
ϕ3,2 3 0 3 3 3 3 2 0 1 1 0 2 0 1 1 2 0 0 2 1 0 3 1 3
ϕ3,8 3 0 3 3 3 3 1 0 2 2 0 1 1 1 0 0 2 2 0 1 0 3 1 3
ϕ3,4 3 0 3 3 3 3 0 1 1 1 0 2 2 0 1 1 1 2 0 0 0 3 2 3
ϕ3,10 3 0 3 3 3 3 1 1 0 2 0 1 1 0 2 1 1 0 2 2 0 3 0 3
ϕ3,6' 3 1 3 3 3 3 2 0 1 1 1 0 2 0 1 0 1 1 1 2 0 3 0 3
ϕ3,6'' 3 0 3 3 3 3 1 0 2 0 1 1 1 0 2 2 0 1 1 0 1 3 1 3
ϕ4,3 4 0 4 4 4 4 1 1 1 2 0 2 1 1 1 1 1 2 2 1 0 4 1 4
ϕ4,5 4 0 4 4 4 4 1 1 1 1 1 1 2 0 2 1 1 1 1 2 0 4 2 4
ϕ4,7 4 0 4 4 4 4 2 0 2 1 1 1 1 1 1 2 2 1 1 1 0 4 1 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 0 t16 t12 t20 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 0 t3 t5 t7
ϕ1,8 t16 1 t8 t28 t12 t20 0 0 t t9 0 0 t5 0 0 0 0 t2 0 0 0 t7 0 t5
ϕ1,16 t8 0 1 t20 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 0 t3
ϕ1,12 t12 0 t28 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 0 t7
ϕ1,20 t28 0 t20 t16 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 t7 0 t5
ϕ1,28 t20 0 t12 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 t5 0 t3
ϕ2,15 t9 + t27 0 t + t19 t15 + t21 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t + t7 t3 0 2t6 0 2t4
ϕ2,12 t12 + t24 0 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 0 t3 + t9 t5 t + t7
ϕ2,9 t15 + t21 0 t7 + t13 t9 + t27 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 0 2t6 t2 2t4
ϕ2,11 t7 + t13 0 t5 + t23 t + t19 t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 0 t2 + t8
ϕ2,8 t4 + t16 0 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 t2 0 0 t + t7 0 2t5
ϕ2,5 t + t19 0 t11 + t17 t7 + t13 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t 2t4 t6 t2 + t8
ϕ2,7 t11 + t17 t t9 + t27 t5 + t23 t7 + t13 t15 + t21 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t t3 0 t5 0 t2 + t8 0 2t6
ϕ2,4 t8 + t20 0 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 0 2t5 t t3 + t9
ϕ2,1 t5 + t23 0 t15 + t21 t11 + t17 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 t3 0 0 t2 + t8 t4 2t6
ϕ3,2 t4 + t10 + t22 0 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 t2 + t8 t4 0 t + 2t7 t3 3t5
ϕ3,8 t10 + t16 + t22 0 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 0 t + 2t7 t3 3t5
ϕ3,4 t2 + t14 + t20 0 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 0 3t5 t + t7 2t3 + t9
ϕ3,10 t8 + t14 + t26 0 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 t4 0 1 + t6 t2 + t8 0 3t5 0 2t3 + t9
ϕ3,6' t6 + t12 + t18 t2 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 t4 t4 1 + t6 0 2t3 + t9 0 t + 2t7
ϕ3,6'' t6 + t18 + t24 0 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 t4 0 1 2t3 + t9 t5 t + 2t7
ϕ4,3 t3 + t9 + t15 + t21 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 t5 t5 t + t7 t + t7 t3 0 1 + 3t6 t2 3t4 + t10
ϕ4,5 t7 + t13 + t19 + t25 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 t5 t5 t + t7 0 3t4 + t10 1 + t6 2t2 + 2t8
ϕ4,7 t5 + t11 + t17 + t23 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 t + t7 t + t7 t3 t3 t5 0 2t2 + 2t8 t4 1 + 3t6

For the generic point of the hyperplane 2k1,1 + k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ3,4,  ϕ3,6'},   {ϕ2,15,  ϕ2,12,  ϕ2,9},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ4,3,  ϕ1,0,  ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,7,  ϕ2,4,  ϕ2,1}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0611 + 2t + 3t2
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,10633 + 2t + t2
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4233 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 5t6 + 4t7
ϕ4,34244 + 5t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,714444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1
ϕ1,8 0 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 1
ϕ1,16 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1
ϕ1,12 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1
ϕ1,20 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 1
ϕ1,28 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1
ϕ2,15 0 2 2 2 2 2 2 0 0 1 0 1 0 0 2 1 0 0 1 1 0 1 2 2
ϕ2,12 0 2 2 2 2 2 0 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 2 2
ϕ2,9 0 2 2 2 2 2 0 0 2 1 0 1 2 0 0 0 1 2 0 0 1 0 2 2
ϕ2,11 0 2 2 2 2 2 2 0 0 2 0 0 1 0 1 0 1 0 0 2 0 1 2 2
ϕ2,8 0 2 2 2 2 2 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 2 2
ϕ2,5 1 2 2 2 2 2 0 0 2 0 0 2 1 0 1 1 0 1 0 0 1 0 2 2
ϕ2,7 0 2 2 2 2 2 1 0 1 2 0 0 2 0 0 0 2 1 0 1 0 0 2 2
ϕ2,4 0 2 2 2 2 2 1 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 2 2
ϕ2,1 0 2 2 2 2 2 1 0 1 0 0 2 0 0 2 2 0 0 0 0 1 1 2 2
ϕ3,2 0 3 3 3 3 3 2 0 1 1 0 2 0 1 1 2 0 0 0 1 1 2 3 3
ϕ3,8 0 3 3 3 3 3 1 0 2 2 0 1 1 1 0 0 2 2 0 1 1 0 3 3
ϕ3,4 1 3 3 3 3 3 0 1 1 1 0 2 2 0 1 1 1 2 0 0 1 0 3 3
ϕ3,10 0 3 3 3 3 3 1 1 0 2 0 1 1 0 2 1 1 0 1 2 0 1 3 3
ϕ3,6' 0 3 3 3 3 3 2 0 1 1 1 0 2 0 1 0 2 1 0 2 0 1 3 3
ϕ3,6'' 0 3 3 3 3 3 1 0 2 0 1 1 1 0 2 2 0 1 0 0 2 1 3 3
ϕ4,3 0 4 4 4 4 4 1 1 1 2 0 2 1 1 1 1 1 2 0 1 1 2 4 4
ϕ4,5 0 4 4 4 4 4 1 1 1 1 1 1 2 0 2 1 1 1 0 2 2 1 4 4
ϕ4,7 0 4 4 4 4 4 2 0 2 1 1 1 1 1 1 2 2 1 0 1 1 1 4 4

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 t20 t28 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 0 0 t5 t7
ϕ1,8 0 1 t8 t28 t12 t20 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 0 t3 t5
ϕ1,16 0 t16 1 t20 t28 t12 t5 0 0 t 0 0 0 0 t9 0 0 0 0 t2 0 t5 t7 t3
ϕ1,12 0 t20 t28 1 t8 t16 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 0 t5 t7
ϕ1,20 0 t12 t20 t16 1 t8 t 0 0 0 0 t9 0 0 t5 t6 0 0 t2 0 0 0 t3 t5
ϕ1,28 0 t28 t12 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 t6 0 0 t2 0 t7 t3
ϕ2,15 0 t11 + t17 t + t19 t15 + t21 t5 + t23 t7 + t13 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 t5 0 0 t t3 0 t6 t2 + t8 2t4
ϕ2,12 0 t8 + t20 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 1 0 t5 0 t5 0 t4 0 t2 t2 0 0 t6 t6 0 2t5 t + t7
ϕ2,9 0 t5 + t23 t7 + t13 t9 + t27 t11 + t17 t + t19 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t + t7 0 0 t3 0 t2 + t8 2t4
ϕ2,11 0 t15 + t21 t5 + t23 t + t19 t9 + t27 t11 + t17 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 0 t + t7 0 t4 2t6 t2 + t8
ϕ2,8 0 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 t4 0 0 1 0 t5 0 t5 t6 t6 t2 0 0 0 t t3 + t9 2t5
ϕ2,5 t t9 + t27 t11 + t17 t7 + t13 t15 + t21 t5 + t23 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 t3 0 t5 0 0 t 0 2t6 t2 + t8
ϕ2,7 0 t + t19 t9 + t27 t5 + t23 t7 + t13 t15 + t21 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 0 2t4 2t6
ϕ2,4 0 t4 + t16 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t5 0 t5 0 t4 0 0 1 0 0 0 t6 0 t2 t2 t5 t + t7 t3 + t9
ϕ2,1 0 t7 + t13 t15 + t21 t11 + t17 t + t19 t9 + t27 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t + t7 0 0 0 0 t5 t2 2t4 2t6
ϕ3,2 0 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 + t6 0 0 0 t4 t4 t + t7 2t3 + t9 3t5
ϕ3,8 0 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 + t8 0 t4 t4 0 2t3 + t9 3t5
ϕ3,4 t2 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 t4 t4 1 + t6 0 0 t2 0 t + 2t7 2t3 + t9
ϕ3,10 0 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 t4 t4 0 1 t2 + t8 0 t5 t + 2t7 2t3 + t9
ϕ3,6' 0 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 t4 0 1 + t6 0 t3 3t5 t + 2t7
ϕ3,6'' 0 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 t2 + t8 0 t4 0 0 1 + t6 t3 3t5 t + 2t7
ϕ4,3 0 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 t5 t5 t + t7 0 t3 t3 1 + t6 2t2 + 2t8 3t4 + t10
ϕ4,5 0 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 t3 t3 t5 0 t + t7 t + t7 t4 1 + 3t6 2t2 + 2t8
ϕ4,7 0 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 t + t7 t + t7 t3 0 t5 t5 t2 3t4 + t10 1 + 3t6

For the generic point of the hyperplane 2k1,1 + 2k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,7,  ϕ3,2,  ϕ1,28,  ϕ3,4,  ϕ3,6'},   {ϕ2,7,  ϕ2,4,  ϕ2,1},   {ϕ3,8,  ϕ3,10,  ϕ3,6''},   {ϕ2,8,  ϕ2,5,  ϕ2,11},   {ϕ2,15,  ϕ2,12,  ϕ2,9}

Dimensions, Poincaré series and diagonal Verma multiplicities of the simple modules

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,814411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1614411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,1214411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,2014411 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 6t12 + 6t13 + 6t14 + 6t15 + 6t16 + 6t17 + 6t18 + 6t19 + 6t20 + 6t21 + 6t22 + 6t23 + 5t24 + 4t25 + 3t26 + 2t27 + t28
ϕ1,28611 + 2t + 3t2
ϕ2,155422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,121842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,95422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,115422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,81842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,55422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,75422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ2,41842 + 4t + 6t2 + 4t3 + 2t4
ϕ2,15422 + 4t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 6t8 + 4t9 + 2t10
ϕ3,2633 + 2t + t2
ϕ3,84833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,44233 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 5t6 + 4t7
ϕ3,104833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6'4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,6''4833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,314444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,514444 + 8t + 12t2 + 16t3 + 20t4 + 24t5 + 20t6 + 16t7 + 12t8 + 8t9 + 4t10
ϕ4,74244 + 5t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 3t7

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 1 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0
ϕ1,8 1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 1 0
ϕ1,16 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 0
ϕ1,12 1 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0
ϕ1,20 1 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1
ϕ1,28 1 1 1 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0
ϕ2,15 2 2 2 2 2 0 2 0 0 1 0 1 0 0 2 0 0 0 2 1 0 2 2 1
ϕ2,12 2 2 2 2 2 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 2 2 1
ϕ2,9 2 2 2 2 2 1 0 0 2 1 0 1 2 0 0 0 1 1 0 0 1 2 2 0
ϕ2,11 2 2 2 2 2 0 2 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 2 0
ϕ2,8 2 2 2 2 2 0 0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 2 2 1
ϕ2,5 2 2 2 2 2 0 0 0 2 0 0 2 1 0 1 0 0 1 0 0 2 2 2 1
ϕ2,7 2 2 2 2 2 0 1 0 1 2 0 0 2 0 0 0 2 1 0 1 0 2 2 0
ϕ2,4 2 2 2 2 2 0 1 0 1 0 1 0 0 1 0 0 0 1 1 1 1 2 2 0
ϕ2,1 2 2 2 2 2 0 1 0 1 0 0 2 0 0 2 1 0 0 1 0 1 2 2 1
ϕ3,2 3 3 3 3 3 0 2 0 1 1 0 2 0 1 1 1 0 0 2 1 1 3 3 1
ϕ3,8 3 3 3 3 3 1 1 0 2 2 0 1 1 1 0 0 2 1 0 1 1 3 3 0
ϕ3,4 3 3 3 3 3 0 0 1 1 1 0 2 2 0 1 0 1 2 0 0 2 3 3 1
ϕ3,10 3 3 3 3 3 0 1 1 0 2 0 1 1 0 2 0 1 0 2 2 0 3 3 1
ϕ3,6' 3 3 3 3 3 0 2 0 1 1 1 0 2 0 1 0 2 1 1 2 0 3 3 0
ϕ3,6'' 3 3 3 3 3 0 1 0 2 0 1 1 1 0 2 0 0 1 1 0 2 3 3 2
ϕ4,3 4 4 4 4 4 0 1 1 1 2 0 2 1 1 1 0 1 2 2 1 1 4 4 1
ϕ4,5 4 4 4 4 4 0 1 1 1 1 1 1 2 0 2 0 1 1 1 2 2 4 4 1
ϕ4,7 4 4 4 4 4 0 2 0 2 1 1 1 1 1 1 0 2 1 1 1 1 4 4 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,8) L(ϕ1,16) L(ϕ1,12) L(ϕ1,20) L(ϕ1,28) L(ϕ2,15) L(ϕ2,12) L(ϕ2,9) L(ϕ2,11) L(ϕ2,8) L(ϕ2,5) L(ϕ2,7) L(ϕ2,4) L(ϕ2,1) L(ϕ3,2) L(ϕ3,8) L(ϕ3,4) L(ϕ3,10) L(ϕ3,6') L(ϕ3,6'') L(ϕ4,3) L(ϕ4,5) L(ϕ4,7)
ϕ1,0 1 t8 t16 t12 t20 0 0 0 t9 0 0 t5 0 0 t t2 0 0 0 0 t6 t3 t5 0
ϕ1,8 t16 1 t8 t28 t12 0 0 0 t t9 0 0 t5 0 0 0 t6 t2 0 0 0 t7 t3 0
ϕ1,16 t8 t16 1 t20 t28 0 t5 0 0 t 0 0 0 0 t9 0 0 0 t6 t2 0 t5 t7 0
ϕ1,12 t12 t20 t28 1 t8 0 t9 0 0 t5 0 0 t 0 0 0 t2 0 0 t6 0 t3 t5 0
ϕ1,20 t28 t12 t20 t16 1 0 t 0 0 0 0 t9 0 0 t5 0 0 0 t2 0 0 t7 t3 t5
ϕ1,28 t20 t28 t12 t8 t16 1 0 0 t5 0 0 t t9 0 0 0 0 0 0 0 t2 t5 t7 0
ϕ2,15 t9 + t27 t11 + t17 t + t19 t15 + t21 t5 + t23 0 1 + t6 0 0 t2 0 t8 0 0 t4 + t10 0 0 0 t + t7 t3 0 2t6 t2 + t8 t4
ϕ2,12 t12 + t24 t8 + t20 t4 + t16 t12 + t24 t8 + t20 0 0 1 0 t5 0 t5 0 t4 0 0 t2 0 0 t6 t6 t3 + t9 2t5 t
ϕ2,9 t15 + t21 t5 + t23 t7 + t13 t9 + t27 t11 + t17 t 0 0 1 + t6 t8 0 t2 t4 + t10 0 0 0 t5 t 0 0 t3 2t6 t2 + t8 0
ϕ2,11 t7 + t13 t15 + t21 t5 + t23 t + t19 t9 + t27 0 t4 + t10 0 0 1 + t6 0 0 t2 0 t8 0 t3 0 t5 t + t7 0 2t4 2t6 0
ϕ2,8 t4 + t16 t12 + t24 t8 + t20 t4 + t16 t12 + t24 0 0 t4 0 0 1 0 t5 0 t5 0 t6 t2 t2 0 0 t + t7 t3 + t9 t5
ϕ2,5 t + t19 t9 + t27 t11 + t17 t7 + t13 t15 + t21 0 0 0 t4 + t10 0 0 1 + t6 t8 0 t2 0 0 t5 0 0 t + t7 2t4 2t6 t2
ϕ2,7 t11 + t17 t + t19 t9 + t27 t5 + t23 t7 + t13 0 t8 0 t2 t4 + t10 0 0 1 + t6 0 0 0 t + t7 t3 0 t5 0 t2 + t8 2t4 0
ϕ2,4 t8 + t20 t4 + t16 t12 + t24 t8 + t20 t4 + t16 0 t5 0 t5 0 t4 0 0 1 0 0 0 t6 t6 t2 t2 2t5 t + t7 0
ϕ2,1 t5 + t23 t7 + t13 t15 + t21 t11 + t17 t + t19 0 t2 0 t8 0 0 t4 + t10 0 0 1 + t6 t 0 0 t3 0 t5 t2 + t8 2t4 t6
ϕ3,2 t4 + t10 + t22 t6 + t12 + t18 t2 + t14 + t20 t10 + t16 + t22 t6 + t18 + t24 0 t + t7 0 t7 t3 0 t3 + t9 0 t2 t5 1 0 0 t2 + t8 t4 t4 t + 2t7 2t3 + t9 t5
ϕ3,8 t10 + t16 + t22 t6 + t18 + t24 t8 + t14 + t26 t4 + t10 + t22 t6 + t12 + t18 t2 t7 0 t + t7 t3 + t9 0 t3 t5 t2 0 0 1 + t6 t2 0 t4 t4 t + 2t7 2t3 + t9 0
ϕ3,4 t2 + t14 + t20 t4 + t10 + t22 t6 + t12 + t18 t8 + t14 + t26 t10 + t16 + t22 0 0 t2 t5 t7 0 t + t7 t3 + t9 0 t3 0 t4 1 + t6 0 0 t2 + t8 3t5 t + 2t7 t3
ϕ3,10 t8 + t14 + t26 t10 + t16 + t22 t6 + t18 + t24 t2 + t14 + t20 t4 + t10 + t22 0 t5 t2 0 t + t7 0 t7 t3 0 t3 + t9 0 t4 0 1 + t6 t2 + t8 0 3t5 t + 2t7 t3
ϕ3,6' t6 + t12 + t18 t2 + t14 + t20 t4 + t10 + t22 t6 + t18 + t24 t8 + t14 + t26 0 t3 + t9 0 t3 t5 t2 0 t + t7 0 t7 0 t2 + t8 t4 t4 1 + t6 0 2t3 + t9 3t5 0
ϕ3,6'' t6 + t18 + t24 t8 + t14 + t26 t10 + t16 + t22 t6 + t12 + t18 t2 + t14 + t20 0 t3 0 t3 + t9 0 t2 t5 t7 0 t + t7 0 0 t4 t4 0 1 + t6 2t3 + t9 3t5 t + t7
ϕ4,3 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 0 t6 t3 t6 t2 + t8 0 t2 + t8 t4 t t4 0 t5 t + t7 t + t7 t3 t3 1 + 3t6 2t2 + 2t8 t4
ϕ4,5 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 0 t4 t t4 t6 t3 t6 t2 + t8 0 t2 + t8 0 t3 t5 t5 t + t7 t + t7 3t4 + t10 1 + 3t6 t2
ϕ4,7 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 t3 + t9 + t15 + t21 t5 + t11 + t17 + t23 t7 + t13 + t19 + t25 0 t2 + t8 0 t2 + t8 t4 t t4 t6 t3 t6 0 t + t7 t3 t3 t5 t5 2t2 + 2t8 3t4 + t10 1 + t6