The representation theory of the restricted rational Cherednik algebra for G8

Computed by Ulrich Thiel using CHAMP (see LMS J. Comput. Math., 2015). Last update on Fri Mar 27 12:48:14 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

{ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
ϕ1,6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
ϕ1,12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
ϕ1,18 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
ϕ2,1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
ϕ2,4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0
ϕ2,7' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
ϕ2,7'' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
ϕ2,10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 2
ϕ2,13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
ϕ3,8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1
ϕ3,6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 2
ϕ3,4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1
ϕ3,2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 2
ϕ4,5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2
ϕ4,3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 t7 t7 t10 t13 t8 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 t10 t13 t t4 t7 t2 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 t6 t13 t4 t7 t7 t10 t t4 t2 t8 t6 0 t3
ϕ1,18 t6 t12 t18 1 t7 t10 t t13 t4 t7 t6 t4 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

Exceptional hyperplanes

k1,3
k1,2
k1,2 − 2k1,3
k1,2 − k1,3
k1,2 + k1,3
2k1,2 − k1,3
k1,1
k1,1 − 2k1,3
k1,1 − k1,3
k1,1 + k1,3
k1,1 − 3k1,2 + k1,3
k1,1 − 2k1,2
k1,1 − 2k1,2 + k1,3
k1,1 − k1,2
k1,1 − k1,2 − k1,3
k1,1 − k1,2 + k1,3
k1,1 + k1,2
k1,1 + k1,2 − 3k1,3
k1,1 + k1,2 − 2k1,3
k1,1 + k1,2 − k1,3
k1,1 + k1,2 + k1,3
2k1,1 − k1,3
2k1,1 − k1,2
2k1,1 − k1,2 − k1,3
3k1,1 − k1,2 − k1,3

For the generic point of the hyperplane k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,8,  ϕ3,6},   {ϕ2,4,  ϕ2,7''},   {ϕ4,5,  ϕ4,3},   {ϕ1,0,  ϕ1,6,  ϕ2,1},   {ϕ2,7',  ϕ2,10}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,04911 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,6111
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,12342 + 3t + 4t2 + 5t3 + 4t4 + 3t5 + 2t6
ϕ2,47222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ2,7'7222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ2,7''2422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ2,102422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,82433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,67233 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 9t6 + 6t7 + 3t8
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1
ϕ1,6 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0
ϕ1,12 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1
ϕ1,18 1 0 1 1 0 1 1 0 0 1 0 1 1 1 1 0
ϕ2,1 0 0 2 2 1 1 1 1 1 2 1 1 2 2 1 1
ϕ2,4 2 0 2 2 0 2 2 0 0 2 0 2 2 2 2 0
ϕ2,7' 2 0 2 2 0 2 2 0 0 2 0 2 2 2 1 1
ϕ2,7'' 0 0 2 2 1 1 1 1 1 2 1 1 2 2 1 1
ϕ2,10 0 0 2 2 1 1 1 1 1 2 1 1 2 2 0 2
ϕ2,13 2 0 2 2 0 2 2 0 0 2 0 2 2 2 1 1
ϕ3,8 1 0 3 3 1 2 2 1 1 3 1 2 3 3 2 1
ϕ3,6 3 0 3 3 0 3 3 0 0 3 0 3 3 3 1 2
ϕ3,4 1 0 3 3 1 2 2 1 1 3 1 2 3 3 2 1
ϕ3,2 1 0 3 3 1 2 2 1 1 3 1 2 3 3 1 2
ϕ4,5 2 0 4 4 1 3 3 1 1 4 1 3 4 4 2 2
ϕ4,3 2 0 4 4 1 3 3 1 1 4 1 3 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 t12 t18 0 t4 t7 0 0 t13 0 t6 t4 t2 0 t3
ϕ1,6 0 1 t6 t12 0 0 0 t t4 t7 t2 0 t6 t4 t3 0
ϕ1,12 t12 0 1 t6 0 t4 t7 0 0 t 0 t2 t8 t6 0 t3
ϕ1,18 t6 0 t18 1 0 t10 t 0 0 t7 0 t4 t2 t8 t3 0
ϕ2,1 0 0 t7 + t11 t13 + t17 1 t3 t6 t2 t5 t8 + t12 t3 t5 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 0 t4 + t8 t10 + t14 0 1 + t8 t3 + t11 0 0 t5 + t9 0 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 0 t13 + t17 t7 + t11 0 t5 + t9 1 + t8 0 0 t6 + t14 0 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' 0 0 t + t5 t7 + t11 t6 t5 t8 1 t3 t2 + t6 t t3 t5 + t9 t3 + t7 t2 t4
ϕ2,10 0 0 t10 + t14 t4 + t8 t3 t6 t5 t5 1 t3 + t11 t2 t4 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 0 t7 + t11 t + t5 0 t3 + t11 t2 + t6 0 0 1 + t8 0 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 0 t4 + t8 + t12 t2 + t6 + t10 t5 t4 + t8 t3 + t7 t3 t2 t + t5 + t9 1 t2 + t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 0 t6 + t10 + t14 t4 + t8 + t12 0 t2 + t6 + t10 t + t5 + t9 0 0 t3 + t7 + t11 0 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 0 t8 + t12 + t16 t6 + t10 + t14 t t4 + t8 t3 + t7 t3 t2 t5 + t9 + t13 t4 t2 + t6 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 0 t2 + t6 + t10 t8 + t12 + t16 t3 t2 + t6 t5 + t9 t t4 t3 + t7 + t11 t2 2t4 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t4 t3 + 2t7 t2 + t6 + t10 t2 t 2t4 + t8 + t12 t3 t + 2t5 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + t9 0 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 t + t5 + t9 2t4 + t8 t4 t3 t2 + t6 + 2t10 t 2t3 + t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,8,  ϕ3,4},   {ϕ1,0,  ϕ1,12,  ϕ2,4},   {ϕ4,5,  ϕ4,3},   {ϕ2,1,  ϕ2,7''},   {ϕ2,7',  ϕ2,13}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,01611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,121611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,14822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ2,43242 + 4t + 6t2 + 8t3 + 6t4 + 4t5 + 2t6
ϕ2,7'4822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ2,7''4822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,134822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ3,84833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 0 1 1 0 1 0 1 0 0 1 1 1 0 1
ϕ1,6 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0
ϕ1,12 0 1 1 1 0 0 0 1 1 1 1 1 0 1 0 1
ϕ1,18 1 1 0 1 1 0 1 0 1 0 0 1 1 1 1 0
ϕ2,1 0 2 0 2 1 1 1 1 2 1 1 2 1 2 1 1
ϕ2,4 0 2 0 2 1 1 1 1 2 1 1 2 1 2 2 0
ϕ2,7' 2 2 0 2 2 0 2 0 2 0 0 2 2 2 1 1
ϕ2,7'' 0 2 2 2 0 0 0 2 2 2 2 2 0 2 1 1
ϕ2,10 0 2 0 2 1 1 1 1 2 1 1 2 1 2 0 2
ϕ2,13 0 2 0 2 1 1 1 1 2 1 1 2 1 2 1 1
ϕ3,8 0 3 1 3 1 1 1 2 3 2 2 3 1 3 2 1
ϕ3,6 1 3 0 3 2 1 2 1 3 1 1 3 2 3 1 2
ϕ3,4 1 3 0 3 2 1 2 1 3 1 1 3 2 3 2 1
ϕ3,2 0 3 1 3 1 1 1 2 3 2 2 3 1 3 1 2
ϕ4,5 1 4 1 4 2 1 2 2 4 2 2 4 2 4 2 2
ϕ4,3 0 4 0 4 2 2 2 2 4 2 2 4 2 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 0 t18 t 0 t7 0 t10 0 0 t6 t4 t2 0 t3
ϕ1,6 0 1 t6 t12 0 0 0 t t4 t7 t2 t8 0 t4 t3 0
ϕ1,12 0 t18 1 t6 0 0 0 t7 t10 t t4 t2 0 t6 0 t3
ϕ1,18 t6 t12 0 1 t7 0 t 0 t4 0 0 t4 t2 t8 t3 0
ϕ2,1 0 t + t5 0 t13 + t17 1 t3 t6 t2 t5 + t9 t8 t3 t5 + t9 t3 t + t5 t4 t2
ϕ2,4 0 t10 + t14 0 t10 + t14 t5 1 t3 t3 t6 + t14 t5 t4 t2 + t6 t4 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 0 t7 + t11 t2 + t6 0 1 + t8 0 t3 + t11 0 0 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' 0 t7 + t11 t + t5 t7 + t11 0 0 0 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 0 t3 + t7 t2 t4
ϕ2,10 0 t4 + t8 0 t4 + t8 t3 t6 t5 t5 1 + t8 t3 t2 t4 + t8 t2 t4 + t8 0 t + t5
ϕ2,13 0 t13 + t17 0 t + t5 t8 t3 t2 t6 t5 + t9 1 t3 t + t5 t3 t5 + t9 t4 t2
ϕ3,8 0 t6 + t10 + t14 t4 t2 + t6 + t10 t5 t4 t3 t3 + t7 t2 + t6 + t10 t + t5 1 + t4 t2 + 2t6 t4 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 t8 + t12 + t16 0 t4 + t8 + t12 t3 + t7 t2 t + t5 t5 t4 + t8 + t12 t3 t2 1 + t4 + t8 t2 + t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 t2 + t6 + t10 0 t6 + t10 + t14 t + t5 t4 t3 + t7 t3 t2 + t6 + t10 t5 t4 t2 + t6 + t10 1 + t4 t2 + 2t6 t + t5 t3
ϕ3,2 0 t4 + t8 + t12 t2 t8 + t12 + t16 t3 t2 t5 t + t5 t4 + t8 + t12 t3 + t7 t2 + t6 2t4 + t8 t2 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 t5 + 2t9 + t13 t3 t5 + 2t9 + t13 2t4 t3 t2 + t6 t2 + t6 t + t5 + t9 + t13 2t4 2t3 t + 2t5 + t9 2t3 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 0 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 t2 + t6 t + t5 2t4 2t4 t3 + 2t7 + t11 t2 + t6 t + t5 2t3 + 2t7 t + t5 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,2 − 2k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,6,  ϕ3,2,  ϕ2,4},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,61211 + 2t + 3t2 + 4t3 + 2t4
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,41222 + 4t + 3t2 + 2t3 + t4
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,27233 + 6t + 7t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 7t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1
ϕ1,6 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0
ϕ1,12 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1
ϕ1,18 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0
ϕ2,1 2 1 2 2 2 0 2 2 2 2 2 2 2 1 1 1
ϕ2,4 2 0 2 2 2 1 2 2 2 2 2 2 2 1 2 0
ϕ2,7' 2 0 2 2 2 0 2 2 2 2 2 2 2 2 1 1
ϕ2,7'' 2 0 2 2 2 0 2 2 2 2 2 2 2 2 1 1
ϕ2,10 2 1 2 2 2 0 2 2 2 2 2 2 2 1 0 2
ϕ2,13 2 0 2 2 2 1 2 2 2 2 2 2 2 1 1 1
ϕ3,8 3 0 3 3 3 0 3 3 3 3 3 3 3 3 2 1
ϕ3,6 3 0 3 3 3 1 3 3 3 3 3 3 3 2 1 2
ϕ3,4 3 1 3 3 3 0 3 3 3 3 3 3 3 2 2 1
ϕ3,2 3 0 3 3 3 0 3 3 3 3 3 3 3 3 1 2
ϕ4,5 4 0 4 4 4 0 4 4 4 4 4 4 4 4 2 2
ϕ4,3 4 1 4 4 4 1 4 4 4 4 4 4 4 2 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 t12 t18 t 0 t7 t7 t10 t13 t8 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 0 t13 t t4 t7 t2 t8 t6 0 t3 0
ϕ1,12 t12 0 1 t6 t13 t4 t7 t7 t10 t t4 t2 t8 0 0 t3
ϕ1,18 t6 0 t18 1 t7 0 t t13 t4 t7 t6 t4 t2 t8 t3 0
ϕ2,1 t7 + t11 t t7 + t11 t13 + t17 1 + t8 0 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t t4 t2
ϕ2,4 t4 + t8 0 t4 + t8 t10 + t14 t5 + t9 1 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t4 + t8 t6 t + t5 0
ϕ2,7' t + t5 0 t13 + t17 t7 + t11 t2 + t6 0 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 0 t + t5 t7 + t11 t6 + t14 0 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 t10 + t14 t4 + t8 t3 + t11 0 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 0 t + t5
ϕ2,13 t7 + t11 0 t7 + t11 t + t5 t8 + t12 t3 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 t9 t4 t2
ϕ3,8 t8 + t12 + t16 0 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 0 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 0 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 0 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 0 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 0 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 0 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 t3 + t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,2 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,10,  ϕ2,13},   {ϕ2,1,  ϕ2,4},   {ϕ1,6,  ϕ1,12,  ϕ2,7''},   {ϕ4,5,  ϕ4,3},   {ϕ3,6,  ϕ3,4}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,64911 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,12111
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,17222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ2,42422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''2342 + 3t + 4t2 + 5t3 + 4t4 + 3t5 + 2t6
ϕ2,107222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ2,132422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,62433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,47233 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 9t6 + 6t7 + 3t8
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 0 1
ϕ1,6 1 1 0 1 1 0 1 0 1 0 1 0 1 1 1 0
ϕ1,12 1 0 1 1 0 1 1 0 0 1 1 1 0 1 0 1
ϕ1,18 1 1 0 1 1 0 1 0 1 0 1 0 1 1 1 0
ϕ2,1 2 2 0 2 2 0 2 0 2 0 2 0 2 2 1 1
ϕ2,4 2 0 0 2 1 1 2 1 1 1 2 1 1 2 2 0
ϕ2,7' 2 2 0 2 2 0 2 0 2 0 2 0 2 2 1 1
ϕ2,7'' 2 0 0 2 1 1 2 1 1 1 2 1 1 2 1 1
ϕ2,10 2 2 0 2 2 0 2 0 2 0 2 0 2 2 0 2
ϕ2,13 2 0 0 2 1 1 2 1 1 1 2 1 1 2 1 1
ϕ3,8 3 1 0 3 2 1 3 1 2 1 3 1 2 3 2 1
ϕ3,6 3 1 0 3 2 1 3 1 2 1 3 1 2 3 1 2
ϕ3,4 3 3 0 3 3 0 3 0 3 0 3 0 3 3 2 1
ϕ3,2 3 1 0 3 2 1 3 1 2 1 3 1 2 3 1 2
ϕ4,5 4 2 0 4 3 1 4 1 3 1 4 1 3 4 2 2
ϕ4,3 4 2 0 4 3 1 4 1 3 1 4 1 3 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 0 t18 t 0 t7 0 t10 0 t8 0 t4 t2 0 t3
ϕ1,6 t18 1 0 t12 t7 0 t13 0 t4 0 t2 0 t6 t4 t3 0
ϕ1,12 t12 0 1 t6 0 t4 t7 0 0 t t4 t2 0 t6 0 t3
ϕ1,18 t6 t12 0 1 t7 0 t 0 t4 0 t6 0 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 0 t13 + t17 1 + t8 0 t6 + t14 0 t5 + t9 0 t3 + t7 0 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 0 0 t10 + t14 t5 1 t3 + t11 t3 t6 t5 t4 + t8 t2 t4 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 0 t7 + t11 t2 + t6 0 1 + t8 0 t3 + t11 0 t5 + t9 0 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 0 0 t7 + t11 t6 t5 t8 + t12 1 t3 t2 t + t5 t3 t5 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 0 t4 + t8 t3 + t11 0 t5 + t9 0 1 + t8 0 t2 + t6 0 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 0 0 t + t5 t8 t3 t2 + t6 t6 t5 1 t3 + t7 t t3 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 0 t2 + t6 + t10 t5 + t9 t4 t3 + t7 + t11 t3 t2 + t6 t 1 + t4 + t8 t2 2t4 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 0 t4 + t8 + t12 t3 + t7 t2 t + t5 + t9 t5 t4 + t8 t3 t2 + t6 + t10 1 t2 + t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 0 t6 + t10 + t14 t + t5 + t9 0 t3 + t7 + t11 0 t2 + t6 + t10 0 2t4 + t8 0 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 0 t8 + t12 + t16 t3 + t7 t2 t5 + t9 + t13 t t4 + t8 t3 t2 + 2t6 t4 t2 + t6 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + t9 0 t5 + 2t9 + t13 2t4 + t8 t3 t2 + t6 + 2t10 t2 t + t5 + t9 t4 2t3 + 2t7 t 2t3 + t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 0 t3 + t7 + t11 + t15 t2 + t6 + t10 t 2t4 + t8 + t12 t4 t3 + 2t7 t2 t + 2t5 + t9 t3 t + 2t5 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,2 + k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ3,2,  ϕ2,7''},   {ϕ4,5,  ϕ4,3}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,02711 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 4t6 + 2t7
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''322 + t
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,26633 + 6t + 8t2 + 8t3 + 8t4 + 8t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1
ϕ1,6 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0
ϕ1,12 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1
ϕ1,18 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0
ϕ2,1 0 2 2 2 2 2 2 0 2 2 2 2 2 2 1 1
ϕ2,4 1 2 2 2 2 2 2 0 2 2 2 2 2 1 2 0
ϕ2,7' 2 2 2 2 2 2 2 0 2 2 2 2 2 0 1 1
ϕ2,7'' 0 2 2 2 2 2 2 1 2 2 2 2 2 1 1 1
ϕ2,10 0 2 2 2 2 2 2 0 2 2 2 2 2 2 0 2
ϕ2,13 1 2 2 2 2 2 2 0 2 2 2 2 2 1 1 1
ϕ3,8 0 3 3 3 3 3 3 0 3 3 3 3 3 3 2 1
ϕ3,6 2 3 3 3 3 3 3 0 3 3 3 3 3 1 1 2
ϕ3,4 1 3 3 3 3 3 3 0 3 3 3 3 3 2 2 1
ϕ3,2 0 3 3 3 3 3 3 0 3 3 3 3 3 3 1 2
ϕ4,5 1 4 4 4 4 4 4 0 4 4 4 4 4 3 2 2
ϕ4,3 1 4 4 4 4 4 4 0 4 4 4 4 4 3 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 t7 0 t10 t13 t8 t6 t4 0 0 t3
ϕ1,6 0 1 t6 t12 t7 t10 t13 t t4 t7 t2 t8 t6 0 t3 0
ϕ1,12 0 t18 1 t6 t13 t4 t7 0 t10 t t4 t2 t8 t6 0 t3
ϕ1,18 t6 t12 t18 1 t7 t10 t 0 t4 t7 t6 t4 t2 0 t3 0
ϕ2,1 0 t + t5 t7 + t11 t13 + t17 1 + t8 t3 + t11 t6 + t14 0 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 t10 + t14 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 + t11 0 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t4 + t8 t2 t + t5 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 + t8 0 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 0 t2 t4
ϕ2,7'' 0 t7 + t11 t + t5 t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 t7 t2 t4
ϕ2,10 0 t4 + t8 t10 + t14 t4 + t8 t3 + t11 t6 + t14 t5 + t9 0 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 t13 + t17 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 0 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 t5 t4 t2
ϕ3,8 0 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 0 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 0 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t4 t3 t + t5
ϕ3,4 t4 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 0 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + t6 t + t5 t3
ϕ3,2 0 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 0 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 0 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 0 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + t7 t2 + t6 1 + t4

For the generic point of the hyperplane 2k1,2 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,12,  ϕ3,2,  ϕ2,1},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,12311 + 2t
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,12722 + 4t + 6t2 + 5t3 + 4t4 + 3t5 + 2t6 + t7
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,26633 + 4t + 5t2 + 6t3 + 7t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1
ϕ1,6 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0
ϕ1,12 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1
ϕ1,18 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0
ϕ2,1 2 2 0 2 1 2 2 2 2 2 2 2 2 1 1 1
ϕ2,4 2 2 0 2 0 2 2 2 2 2 2 2 2 2 2 0
ϕ2,7' 2 2 0 2 2 2 2 2 2 2 2 2 2 0 1 1
ϕ2,7'' 2 2 1 2 0 2 2 2 2 2 2 2 2 1 1 1
ϕ2,10 2 2 0 2 1 2 2 2 2 2 2 2 2 1 0 2
ϕ2,13 2 2 0 2 0 2 2 2 2 2 2 2 2 2 1 1
ϕ3,8 3 3 0 3 0 3 3 3 3 3 3 3 3 3 2 1
ϕ3,6 3 3 0 3 1 3 3 3 3 3 3 3 3 2 1 2
ϕ3,4 3 3 0 3 2 3 3 3 3 3 3 3 3 1 2 1
ϕ3,2 3 3 0 3 0 3 3 3 3 3 3 3 3 3 1 2
ϕ4,5 4 4 0 4 1 4 4 4 4 4 4 4 4 3 2 2
ϕ4,3 4 4 0 4 1 4 4 4 4 4 4 4 4 3 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 0 t18 t t4 t7 t7 t10 t13 t8 t6 t4 0 0 t3
ϕ1,6 t18 1 0 t12 0 t10 t13 t t4 t7 t2 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 t6 0 t4 t7 t7 t10 t t4 t2 t8 0 0 t3
ϕ1,18 t6 t12 0 1 t7 t10 t t13 t4 t7 t6 t4 t2 0 t3 0
ϕ2,1 t7 + t11 t + t5 0 t13 + t17 1 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 0 t10 + t14 0 1 + t8 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 0 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 0 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t t7 + t11 0 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 t3 t2 t4
ϕ2,10 t10 + t14 t4 + t8 0 t4 + t8 t3 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 0 t + t5 0 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 0 t2 + t6 + t10 0 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 0 t4 + t8 + t12 t3 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 0 t6 + t10 + t14 t + t5 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 0 t8 + t12 + t16 0 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 0 t5 + 2t9 + t13 t4 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 t2 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,4,  ϕ2,13},   {ϕ1,0,  ϕ1,18,  ϕ2,7'},   {ϕ2,1,  ϕ2,10},   {ϕ4,5,  ϕ4,3},   {ϕ3,8,  ϕ3,2}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0111
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,184911 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ2,12422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ2,42422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ2,7'2342 + 3t + 4t2 + 5t3 + 4t4 + 3t5 + 2t6
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,107222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ2,137222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ3,87233 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 9t6 + 6t7 + 3t8
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 1
ϕ1,6 0 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0
ϕ1,12 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1
ϕ1,18 0 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0
ϕ2,1 0 2 2 0 1 1 1 2 1 1 1 2 2 1 1 1
ϕ2,4 0 2 2 0 1 1 1 2 1 1 1 2 2 1 2 0
ϕ2,7' 0 2 2 0 1 1 1 2 1 1 1 2 2 1 1 1
ϕ2,7'' 0 2 2 2 0 0 0 2 2 2 2 2 2 0 1 1
ϕ2,10 0 2 2 2 0 0 0 2 2 2 2 2 2 0 0 2
ϕ2,13 0 2 2 2 0 0 0 2 2 2 2 2 2 0 1 1
ϕ3,8 0 3 3 3 0 0 0 3 3 3 3 3 3 0 2 1
ϕ3,6 0 3 3 1 1 1 1 3 2 2 2 3 3 1 1 2
ϕ3,4 0 3 3 1 1 1 1 3 2 2 2 3 3 1 2 1
ϕ3,2 0 3 3 1 1 1 1 3 2 2 2 3 3 1 1 2
ϕ4,5 0 4 4 2 1 1 1 4 3 3 3 4 4 1 2 2
ϕ4,3 0 4 4 2 1 1 1 4 3 3 3 4 4 1 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 0 t t4 0 t7 0 0 0 t6 t4 t2 0 t3
ϕ1,6 0 1 t6 t12 0 0 0 t t4 t7 t2 t8 t6 0 t3 0
ϕ1,12 0 t18 1 t6 0 0 0 t7 t10 t t4 t2 t8 0 0 t3
ϕ1,18 0 t12 t18 1 0 0 0 t13 t4 t7 t6 t4 t2 0 t3 0
ϕ2,1 0 t + t5 t7 + t11 0 1 t3 t6 t2 + t6 t5 t8 t3 t5 + t9 t3 + t7 t t4 t2
ϕ2,4 0 t10 + t14 t4 + t8 0 t5 1 t3 t3 + t11 t6 t5 t4 t2 + t6 t4 + t8 t2 t + t5 0
ϕ2,7' 0 t7 + t11 t13 + t17 0 t2 t5 1 t8 + t12 t3 t6 t5 t3 + t7 t + t5 t3 t2 t4
ϕ2,7'' 0 t7 + t11 t + t5 t7 + t11 0 0 0 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 0 t2 t4
ϕ2,10 0 t4 + t8 t10 + t14 t4 + t8 0 0 0 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 0 0 t + t5
ϕ2,13 0 t13 + t17 t7 + t11 t + t5 0 0 0 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 0 t4 t2
ϕ3,8 0 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 0 0 0 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 0 t + t5 t3
ϕ3,6 0 t8 + t12 + t16 t6 + t10 + t14 t4 t3 t2 t t5 + t9 + t13 t4 + t8 t3 + t7 t2 + t6 1 + t4 + t8 t2 + 2t6 t4 t3 t + t5
ϕ3,4 0 t2 + t6 + t10 t8 + t12 + t16 t6 t t4 t3 t3 + t7 + t11 t2 + t6 t5 + t9 2t4 t2 + t6 + t10 1 + t4 + t8 t2 t + t5 t3
ϕ3,2 0 t4 + t8 + t12 t2 + t6 + t10 t8 t3 t2 t5 t + t5 + t9 t4 + t8 t3 + t7 t2 + t6 2t4 + t8 t2 + t6 + t10 1 t3 t + t5
ϕ4,5 0 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + t9 t4 t3 t2 t2 + t6 + 2t10 t + t5 + t9 2t4 + t8 2t3 + t7 t + 2t5 + t9 2t3 + 2t7 t 1 + t4 t2 + t6
ϕ4,3 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 t2 t t4 2t4 + t8 + t12 t3 + 2t7 t2 + t6 + t10 t + 2t5 2t3 + 2t7 t + 2t5 + t9 t3 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 − 2k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,5,  ϕ4,3},   {ϕ1,6,  ϕ3,4,  ϕ2,7'}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,6311 + 2t
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'2722 + 4t + 6t2 + 5t3 + 4t4 + 3t5 + 2t6 + t7
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,46633 + 4t + 5t2 + 6t3 + 7t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1
ϕ1,6 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0
ϕ1,12 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 1
ϕ1,18 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0
ϕ2,1 2 1 2 2 2 2 0 2 2 2 2 2 1 2 1 1
ϕ2,4 2 0 2 2 2 2 1 2 2 2 2 2 1 2 2 0
ϕ2,7' 2 0 2 2 2 2 1 2 2 2 2 2 1 2 1 1
ϕ2,7'' 2 0 2 2 2 2 0 2 2 2 2 2 2 2 1 1
ϕ2,10 2 0 2 2 2 2 0 2 2 2 2 2 2 2 0 2
ϕ2,13 2 0 2 2 2 2 2 2 2 2 2 2 0 2 1 1
ϕ3,8 3 0 3 3 3 3 1 3 3 3 3 3 2 3 2 1
ϕ3,6 3 0 3 3 3 3 2 3 3 3 3 3 1 3 1 2
ϕ3,4 3 0 3 3 3 3 0 3 3 3 3 3 3 3 2 1
ϕ3,2 3 0 3 3 3 3 0 3 3 3 3 3 3 3 1 2
ϕ4,5 4 0 4 4 4 4 1 4 4 4 4 4 3 4 2 2
ϕ4,3 4 0 4 4 4 4 1 4 4 4 4 4 3 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 t12 t18 t t4 0 t7 t10 t13 t8 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 t10 0 t t4 t7 t2 t8 0 t4 t3 0
ϕ1,12 t12 0 1 t6 t13 t4 t7 t7 t10 t t4 t2 0 t6 0 t3
ϕ1,18 t6 0 t18 1 t7 t10 t t13 t4 t7 t6 t4 0 t8 t3 0
ϕ2,1 t7 + t11 t t7 + t11 t13 + t17 1 + t8 t3 + t11 0 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t3 t + t5 t4 t2
ϕ2,4 t4 + t8 0 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 0 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 0 t + t5 t7 + t11 t6 + t14 t5 + t9 0 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 0 t10 + t14 t4 + t8 t3 + t11 t6 + t14 0 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 0 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 0 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 0 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 0 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 0 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 0 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 0 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 0 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 0 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 t4 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,6,  ϕ1,18,  ϕ2,10},   {ϕ4,5,  ϕ4,3},   {ϕ3,6,  ϕ3,2},   {ϕ2,7'',  ϕ2,13},   {ϕ2,1,  ϕ2,7'}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,61611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,181611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ2,14822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'4822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ2,7''4822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ2,103242 + 4t + 6t2 + 8t3 + 6t4 + 4t5 + 2t6
ϕ2,134822 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 6t6 + 4t7 + 2t8
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,64833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1
ϕ1,6 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 0
ϕ1,12 1 0 1 1 0 1 1 0 0 1 1 1 1 0 0 1
ϕ1,18 1 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0
ϕ2,1 2 2 2 0 2 2 0 2 0 0 2 0 2 2 1 1
ϕ2,4 2 0 2 0 1 2 1 1 1 1 2 1 2 1 2 0
ϕ2,7' 2 0 2 0 1 2 1 1 1 1 2 1 2 1 1 1
ϕ2,7'' 2 0 2 0 1 2 1 1 1 1 2 1 2 1 1 1
ϕ2,10 2 0 2 0 1 2 1 1 1 1 2 1 2 1 0 2
ϕ2,13 2 0 2 2 0 2 2 0 0 2 2 2 2 0 1 1
ϕ3,8 3 0 3 1 1 3 2 1 1 2 3 2 3 1 2 1
ϕ3,6 3 0 3 1 1 3 2 1 1 2 3 2 3 1 1 2
ϕ3,4 3 1 3 0 2 3 1 2 1 1 3 1 3 2 2 1
ϕ3,2 3 1 3 0 2 3 1 2 1 1 3 1 3 2 1 2
ϕ4,5 4 0 4 0 2 4 2 2 2 2 4 2 4 2 2 2
ϕ4,3 4 1 4 1 2 4 2 2 1 2 4 2 4 2 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 0 t t4 0 t7 0 0 t8 0 t4 t2 0 t3
ϕ1,6 t18 1 t6 0 t7 t10 0 t 0 0 t2 0 t6 t4 t3 0
ϕ1,12 t12 0 1 t6 0 t4 t7 0 0 t t4 t2 t8 0 0 t3
ϕ1,18 t6 0 t18 1 0 t10 t 0 0 t7 t6 t4 t2 0 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 0 1 + t8 t3 + t11 0 t2 + t6 0 0 t3 + t7 0 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 0 t4 + t8 0 t5 1 + t8 t3 t3 t6 t5 t4 + t8 t2 t4 + t8 t2 t + t5 0
ϕ2,7' t + t5 0 t13 + t17 0 t2 t5 + t9 1 t8 t3 t6 t5 + t9 t3 t + t5 t3 t2 t4
ϕ2,7'' t13 + t17 0 t + t5 0 t6 t5 + t9 t8 1 t3 t2 t + t5 t3 t5 + t9 t3 t2 t4
ϕ2,10 t10 + t14 0 t10 + t14 0 t3 t6 + t14 t5 t5 1 t3 t2 + t6 t4 t2 + t6 t4 0 t + t5
ϕ2,13 t7 + t11 0 t7 + t11 t + t5 0 t3 + t11 t2 + t6 0 0 1 + t8 t3 + t7 t + t5 t3 + t7 0 t4 t2
ϕ3,8 t8 + t12 + t16 0 t4 + t8 + t12 t2 t5 t4 + t8 + t12 t3 + t7 t3 t2 t + t5 1 + t4 + t8 t2 + t6 2t4 + t8 t2 t + t5 t3
ϕ3,6 t2 + t6 + t10 0 t6 + t10 + t14 t4 t3 t2 + t6 + t10 t + t5 t5 t4 t3 + t7 t2 + t6 + t10 1 + t4 t2 + 2t6 t4 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 t8 + t12 + t16 0 t + t5 t4 + t8 + t12 t3 t3 + t7 t2 t5 2t4 + t8 t2 1 + t4 + t8 t2 + t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 t2 + t6 + t10 0 t3 + t7 t2 + t6 + t10 t5 t + t5 t4 t3 t2 + 2t6 t4 t2 + t6 + t10 1 + t4 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 0 2t4 t3 + 2t7 + t11 t2 + t6 t2 + t6 t + t5 2t4 2t3 + 2t7 t + t5 2t3 + 2t7 t + t5 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 t5 + 2t9 + t13 t3 t2 + t6 t + t5 + t9 + t13 2t4 2t4 t3 t2 + t6 t + 2t5 + t9 2t3 t + 2t5 + t9 2t3 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 + k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ3,4,  ϕ2,10},   {ϕ4,5,  ϕ4,3}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,01211 + 2t + 3t2 + 4t3 + 2t4
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,101222 + 4t + 3t2 + 2t3 + t4
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,47233 + 6t + 7t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 7t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1
ϕ1,6 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0
ϕ1,12 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1
ϕ1,18 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0
ϕ2,1 0 2 2 2 2 2 2 2 0 2 2 2 2 2 1 1
ϕ2,4 1 2 2 2 2 2 2 2 0 2 2 2 1 2 2 0
ϕ2,7' 1 2 2 2 2 2 2 2 0 2 2 2 1 2 1 1
ϕ2,7'' 0 2 2 2 2 2 2 2 1 2 2 2 1 2 1 1
ϕ2,10 0 2 2 2 2 2 2 2 1 2 2 2 1 2 0 2
ϕ2,13 0 2 2 2 2 2 2 2 0 2 2 2 2 2 1 1
ϕ3,8 0 3 3 3 3 3 3 3 1 3 3 3 2 3 2 1
ϕ3,6 1 3 3 3 3 3 3 3 0 3 3 3 2 3 1 2
ϕ3,4 0 3 3 3 3 3 3 3 0 3 3 3 3 3 2 1
ϕ3,2 0 3 3 3 3 3 3 3 0 3 3 3 3 3 1 2
ϕ4,5 1 4 4 4 4 4 4 4 1 4 4 4 2 4 2 2
ϕ4,3 0 4 4 4 4 4 4 4 0 4 4 4 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 t7 t7 0 t13 t8 t6 0 t2 0 t3
ϕ1,6 0 1 t6 t12 t7 t10 t13 t t4 t7 t2 t8 0 t4 t3 0
ϕ1,12 0 t18 1 t6 t13 t4 t7 t7 0 t t4 t2 t8 t6 0 t3
ϕ1,18 0 t12 t18 1 t7 t10 t t13 0 t7 t6 t4 t2 t8 t3 0
ϕ2,1 0 t + t5 t7 + t11 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 0 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 t10 + t14 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 0 t5 + t9 t4 + t8 t2 + t6 t4 t2 + t6 t + t5 0
ϕ2,7' t t7 + t11 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 0 t6 + t14 t5 + t9 t3 + t7 t t3 + t7 t2 t4
ϕ2,7'' 0 t7 + t11 t + t5 t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 t2 + t6 t + t5 t3 + t7 t9 t3 + t7 t2 t4
ϕ2,10 0 t4 + t8 t10 + t14 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 t3 + t11 t2 + t6 t4 + t8 t6 t4 + t8 0 t + t5
ϕ2,13 0 t13 + t17 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 0 1 + t8 t3 + t7 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 0 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 t + t5 + t9 1 + t4 + t8 t2 + 2t6 t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 0 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + t6 2t4 + t8 t3 t + t5
ϕ3,4 0 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 0 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 0 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 0 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 t3 + t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 0 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 − 3k1,2 + k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,4,  ϕ1,12,  ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,12611 + 2t + 3t2
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,4633 + 2t + t2
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54244 + 5t + 6t2 + 7t3 + 8t4 + 8t5 + 4t6
ϕ4,34244 + 8t + 8t2 + 7t3 + 6t4 + 5t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1
ϕ1,6 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0
ϕ1,12 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0
ϕ1,18 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0
ϕ2,1 2 2 0 2 2 2 2 2 2 2 2 2 0 2 1 1
ϕ2,4 2 2 0 2 2 2 2 2 2 2 2 2 0 2 2 0
ϕ2,7' 2 2 0 2 2 2 2 2 2 2 2 2 1 2 0 1
ϕ2,7'' 2 2 1 2 2 2 2 2 2 2 2 2 0 2 1 0
ϕ2,10 2 2 0 2 2 2 2 2 2 2 2 2 0 2 0 2
ϕ2,13 2 2 0 2 2 2 2 2 2 2 2 2 0 2 1 1
ϕ3,8 3 3 0 3 3 3 3 3 3 3 3 3 0 3 2 1
ϕ3,6 3 3 0 3 3 3 3 3 3 3 3 3 0 3 1 2
ϕ3,4 3 3 0 3 3 3 3 3 3 3 3 3 1 3 1 1
ϕ3,2 3 3 1 3 3 3 3 3 3 3 3 3 0 3 1 1
ϕ4,5 4 4 0 4 4 4 4 4 4 4 4 4 0 4 2 2
ϕ4,3 4 4 0 4 4 4 4 4 4 4 4 4 0 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 0 t18 t t4 t7 t7 t10 t13 t8 t6 0 t2 0 t3
ϕ1,6 t18 1 0 t12 t7 t10 t13 t t4 t7 t2 t8 0 t4 t3 0
ϕ1,12 t12 t18 1 t6 t13 t4 t7 t7 t10 t t4 t2 0 t6 0 0
ϕ1,18 t6 t12 0 1 t7 t10 t t13 t4 t7 t6 t4 t2 t8 0 0
ϕ2,1 t7 + t11 t + t5 0 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 + t9 0 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 0 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 + t6 0 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 0 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t t3 + t7 0 t4
ϕ2,7'' t13 + t17 t7 + t11 t t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 0 t3 + t7 t2 0
ϕ2,10 t10 + t14 t4 + t8 0 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 0 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 0 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 0 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 0 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 0 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 0 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 0 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 0 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 t2 + 2t6 t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 0 1 + t4 + t8 t3 t
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 0 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 0 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 0 2t3 + 2t7 t2 + t6 1 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,6,  ϕ1,12,  ϕ2,7'},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,122711 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 4t6 + 2t7
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'322 + t
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,66633 + 6t + 8t2 + 8t3 + 8t4 + 8t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1
ϕ1,6 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 0
ϕ1,12 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1
ϕ1,18 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0
ϕ2,1 2 2 1 2 2 2 0 2 2 2 2 1 2 2 1 1
ϕ2,4 2 2 1 2 2 2 0 2 2 2 2 1 2 2 2 0
ϕ2,7' 2 2 0 2 2 2 1 2 2 2 2 1 2 2 1 1
ϕ2,7'' 2 2 2 2 2 2 0 2 2 2 2 0 2 2 1 1
ϕ2,10 2 2 0 2 2 2 0 2 2 2 2 2 2 2 0 2
ϕ2,13 2 2 0 2 2 2 0 2 2 2 2 2 2 2 1 1
ϕ3,8 3 3 1 3 3 3 0 3 3 3 3 2 3 3 2 1
ϕ3,6 3 3 0 3 3 3 0 3 3 3 3 3 3 3 1 2
ϕ3,4 3 3 0 3 3 3 0 3 3 3 3 3 3 3 2 1
ϕ3,2 3 3 2 3 3 3 0 3 3 3 3 1 3 3 1 2
ϕ4,5 4 4 1 4 4 4 0 4 4 4 4 3 4 4 2 2
ϕ4,3 4 4 1 4 4 4 0 4 4 4 4 3 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 0 t18 t t4 0 t7 t10 t13 t8 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 t10 0 t t4 t7 t2 0 t6 t4 t3 0
ϕ1,12 t12 t18 1 t6 t13 t4 0 t7 t10 t t4 0 t8 t6 0 t3
ϕ1,18 t6 t12 0 1 t7 t10 t t13 t4 t7 t6 0 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 t13 + t17 1 + t8 t3 + t11 0 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 t10 + t14 t5 + t9 1 + t8 0 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 0 t7 + t11 t2 + t6 t5 + t9 1 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 t7 + t11 t6 + t14 t5 + t9 0 1 + t8 t3 + t11 t2 + t6 t + t5 0 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 0 t4 + t8 t3 + t11 t6 + t14 0 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 0 t + t5 t8 + t12 t3 + t11 0 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 0 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 0 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 0 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 0 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 0 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 0 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 t4 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 0 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 0 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 − 2k1,2 + k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,8,  ϕ1,12,  ϕ2,10},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,121211 + 2t + 3t2 + 4t3 + 2t4
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,101222 + 4t + 3t2 + 2t3 + t4
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,87233 + 6t + 7t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 7t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1
ϕ1,6 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0
ϕ1,12 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1
ϕ1,18 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 0
ϕ2,1 2 2 0 2 2 2 2 2 0 2 2 2 2 2 1 1
ϕ2,4 2 2 1 2 2 2 2 2 0 2 1 2 2 2 2 0
ϕ2,7' 2 2 0 2 2 2 2 2 1 2 1 2 2 2 1 1
ϕ2,7'' 2 2 1 2 2 2 2 2 0 2 1 2 2 2 1 1
ϕ2,10 2 2 0 2 2 2 2 2 1 2 1 2 2 2 0 2
ϕ2,13 2 2 0 2 2 2 2 2 0 2 2 2 2 2 1 1
ϕ3,8 3 3 0 3 3 3 3 3 0 3 3 3 3 3 2 1
ϕ3,6 3 3 0 3 3 3 3 3 0 3 3 3 3 3 1 2
ϕ3,4 3 3 0 3 3 3 3 3 1 3 2 3 3 3 2 1
ϕ3,2 3 3 1 3 3 3 3 3 0 3 2 3 3 3 1 2
ϕ4,5 4 4 1 4 4 4 4 4 1 4 2 4 4 4 2 2
ϕ4,3 4 4 0 4 4 4 4 4 0 4 4 4 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 0 t18 t t4 t7 t7 0 t13 t8 t6 t4 t2 0 t3
ϕ1,6 t18 1 0 t12 t7 t10 t13 t 0 t7 t2 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 t6 t13 t4 t7 t7 0 t 0 t2 t8 t6 0 t3
ϕ1,18 t6 t12 0 1 t7 t10 t t13 t4 t7 0 t4 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 0 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 0 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 0 t5 + t9 t4 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 0 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 t6 + t14 t9 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 0 t2 + t6 t t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 0 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 t3 + t11 t6 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 0 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 0 1 + t8 t3 + t7 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 0 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 0 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 0 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 0 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 0 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 t5 + t9 + t13 t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 0 t3 + t7 + t11 t2 + t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t 2t4 + t8 + t12 t3 + t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 0 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,4,  ϕ3,2},   {ϕ1,12,  ϕ1,18,  ϕ2,13},   {ϕ2,4,  ϕ2,7'},   {ϕ4,5,  ϕ4,3},   {ϕ2,7'',  ϕ2,10}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,124911 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,18111
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,47222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ2,7'2422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ2,7''7222 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 4t10 + 2t11
ϕ2,102422 + 4t + 6t2 + 6t3 + 4t4 + 2t5
ϕ2,132342 + 3t + 4t2 + 5t3 + 4t4 + 3t5 + 2t6
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,42433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,27233 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 9t6 + 6t7 + 3t8
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1
ϕ1,6 1 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0
ϕ1,12 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1
ϕ1,18 1 1 0 1 1 0 1 0 1 0 1 1 1 0 1 0
ϕ2,1 2 2 2 0 2 2 0 2 0 0 2 2 0 2 1 1
ϕ2,4 2 2 2 0 2 2 0 2 0 0 2 2 0 2 2 0
ϕ2,7' 2 2 0 0 2 1 1 1 1 1 2 2 1 1 1 1
ϕ2,7'' 2 2 2 0 2 2 0 2 0 0 2 2 0 2 1 1
ϕ2,10 2 2 0 0 2 1 1 1 1 1 2 2 1 1 0 2
ϕ2,13 2 2 0 0 2 1 1 1 1 1 2 2 1 1 1 1
ϕ3,8 3 3 1 0 3 2 1 2 1 1 3 3 1 2 2 1
ϕ3,6 3 3 1 0 3 2 1 2 1 1 3 3 1 2 1 2
ϕ3,4 3 3 1 0 3 2 1 2 1 1 3 3 1 2 2 1
ϕ3,2 3 3 3 0 3 3 0 3 0 0 3 3 0 3 1 2
ϕ4,5 4 4 2 0 4 3 1 3 1 1 4 4 1 3 2 2
ϕ4,3 4 4 2 0 4 3 1 3 1 1 4 4 1 3 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 0 t t4 0 t7 0 0 t8 t6 0 t2 0 t3
ϕ1,6 t18 1 t6 0 t7 t10 0 t 0 0 t2 t8 0 t4 t3 0
ϕ1,12 t12 t18 1 0 t13 t4 0 t7 0 0 t4 t2 0 t6 0 t3
ϕ1,18 t6 t12 0 1 t7 0 t 0 t4 0 t6 t4 t2 0 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 0 1 + t8 t3 + t11 0 t2 + t6 0 0 t3 + t7 t5 + t9 0 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 0 t5 + t9 1 + t8 0 t3 + t11 0 0 t4 + t8 t2 + t6 0 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 0 0 t2 + t6 t5 1 t8 t3 t6 t5 + t9 t3 + t7 t t3 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 0 t6 + t14 t5 + t9 0 1 + t8 0 0 t + t5 t3 + t7 0 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 0 0 t3 + t11 t6 t5 t5 1 t3 t2 + t6 t4 + t8 t2 t4 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 0 0 t8 + t12 t3 t2 t6 t5 1 t3 + t7 t + t5 t3 t5 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 0 t5 + t9 + t13 t4 + t8 t3 t3 + t7 t2 t 1 + t4 + t8 t2 + 2t6 t4 t2 + t6 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 0 t3 + t7 + t11 t2 + t6 t t5 + t9 t4 t3 t2 + t6 + t10 1 + t4 + t8 t2 2t4 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 0 t + t5 + t9 t4 + t8 t3 t3 + t7 t2 t5 2t4 + t8 t2 + t6 + t10 1 t2 + t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 0 t3 + t7 + t11 t2 + t6 + t10 0 t + t5 + t9 0 0 t2 + 2t6 2t4 + t8 0 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 0 2t4 + t8 + t12 t3 + 2t7 t2 t2 + t6 + t10 t t4 2t3 + 2t7 t + 2t5 + t9 t3 t + 2t5 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + t9 0 t2 + t6 + 2t10 t + t5 + t9 t4 2t4 + t8 t3 t2 t + 2t5 + t9 2t3 + 2t7 t 2t3 + t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 − k1,2 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,5,  ϕ4,3,  ϕ2,7',  ϕ2,7''}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'822 + 4t + 2t2
ϕ2,7''822 + 4t + 2t2
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,51684 + 8t + 4t2
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1
ϕ1,6 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0
ϕ1,12 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1
ϕ1,18 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0
ϕ2,1 2 2 2 2 2 2 0 1 2 2 2 2 2 2 0 1
ϕ2,4 2 2 2 2 2 2 0 0 2 2 2 2 2 2 1 0
ϕ2,7' 2 2 2 2 2 2 1 0 2 2 2 2 2 2 0 1
ϕ2,7'' 2 2 2 2 2 2 0 1 2 2 2 2 2 2 0 1
ϕ2,10 2 2 2 2 2 2 0 0 2 2 2 2 2 2 0 2
ϕ2,13 2 2 2 2 2 2 1 0 2 2 2 2 2 2 0 1
ϕ3,8 3 3 3 3 3 3 0 0 3 3 3 3 3 3 1 1
ϕ3,6 3 3 3 3 3 3 1 0 3 3 3 3 3 3 0 2
ϕ3,4 3 3 3 3 3 3 0 0 3 3 3 3 3 3 1 1
ϕ3,2 3 3 3 3 3 3 0 1 3 3 3 3 3 3 0 2
ϕ4,5 4 4 4 4 4 4 0 0 4 4 4 4 4 4 1 2
ϕ4,3 4 4 4 4 4 4 0 0 4 4 4 4 4 4 1 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 0 0 t10 t13 t8 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 t10 0 t t4 t7 t2 t8 t6 t4 0 0
ϕ1,12 t12 t18 1 t6 t13 t4 0 0 t10 t t4 t2 t8 t6 0 t3
ϕ1,18 t6 t12 t18 1 t7 t10 t 0 t4 t7 t6 t4 t2 t8 0 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 t13 + t17 1 + t8 t3 + t11 0 t2 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t + t5 0 t2
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 t10 + t14 t5 + t9 1 + t8 0 0 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t4 + t8 t2 + t6 t 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 0 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 t3 + t7 0 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 t7 + t11 t6 + t14 t5 + t9 0 1 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 t3 + t7 0 t4
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 t4 + t8 t3 + t11 t6 + t14 0 0 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 0 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 t5 + t9 0 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 0 0 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t 0 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 0 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 0 0 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 0 t t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 0 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 0 0 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 0 0 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 1 + t4

For the generic point of the hyperplane k1,1 − k1,2 + k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,5,  ϕ4,3,  ϕ2,4,  ϕ2,10}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,4242
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,10242
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54444 + 6t + 8t2 + 8t3 + 8t4 + 6t5 + 4t6
ϕ4,34444 + 6t + 8t2 + 8t3 + 8t4 + 6t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1
ϕ1,6 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0
ϕ1,12 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1
ϕ1,18 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0
ϕ2,1 2 2 2 2 2 0 2 2 0 2 2 2 2 2 1 1
ϕ2,4 2 2 2 2 2 1 2 2 0 2 2 2 2 2 0 0
ϕ2,7' 2 2 2 2 2 0 2 2 0 2 2 2 2 2 1 1
ϕ2,7'' 2 2 2 2 2 0 2 2 0 2 2 2 2 2 1 1
ϕ2,10 2 2 2 2 2 0 2 2 1 2 2 2 2 2 0 0
ϕ2,13 2 2 2 2 2 0 2 2 0 2 2 2 2 2 1 1
ϕ3,8 3 3 3 3 3 0 3 3 0 3 3 3 3 3 2 1
ϕ3,6 3 3 3 3 3 0 3 3 0 3 3 3 3 3 1 2
ϕ3,4 3 3 3 3 3 0 3 3 0 3 3 3 3 3 2 1
ϕ3,2 3 3 3 3 3 0 3 3 0 3 3 3 3 3 1 2
ϕ4,5 4 4 4 4 4 0 4 4 0 4 4 4 4 4 2 2
ϕ4,3 4 4 4 4 4 0 4 4 0 4 4 4 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t 0 t7 t7 0 t13 t8 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 0 t13 t 0 t7 t2 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 t6 t13 0 t7 t7 0 t t4 t2 t8 t6 0 t3
ϕ1,18 t6 t12 t18 1 t7 0 t t13 0 t7 t6 t4 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 t13 + t17 1 + t8 0 t6 + t14 t2 + t6 0 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 t10 + t14 t5 + t9 1 t3 + t11 t3 + t11 0 t5 + t9 t4 + t8 t2 + t6 t4 + t8 t2 + t6 0 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 t7 + t11 t2 + t6 0 1 + t8 t8 + t12 0 t6 + t14 t5 + t9 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 t7 + t11 t6 + t14 0 t8 + t12 1 + t8 0 t2 + t6 t + t5 t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 t4 + t8 t3 + t11 0 t5 + t9 t5 + t9 1 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 0
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t + t5 t8 + t12 0 t2 + t6 t6 + t14 0 1 + t8 t3 + t7 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 0 t3 + t7 + t11 t3 + t7 + t11 0 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 0 t + t5 + t9 t5 + t9 + t13 0 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 0 t3 + t7 + t11 t3 + t7 + t11 0 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 0 t5 + t9 + t13 t + t5 + t9 0 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 0 t2 + t6 + 2t10 t2 + t6 + 2t10 0 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 0 2t4 + t8 + t12 2t4 + t8 + t12 0 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,5,  ϕ4,3},   {ϕ1,0,  ϕ3,6,  ϕ2,13}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0311 + 2t
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,132722 + 4t + 6t2 + 5t3 + 4t4 + 3t5 + 2t6 + t7
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,66633 + 4t + 5t2 + 6t3 + 7t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1
ϕ1,6 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0
ϕ1,12 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1
ϕ1,18 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0
ϕ2,1 0 2 2 2 2 2 2 2 2 0 2 2 2 2 1 1
ϕ2,4 0 2 2 2 2 2 2 2 2 0 2 2 2 2 2 0
ϕ2,7' 1 2 2 2 2 2 2 2 2 0 2 1 2 2 1 1
ϕ2,7'' 0 2 2 2 2 2 2 2 2 2 2 0 2 2 1 1
ϕ2,10 0 2 2 2 2 2 2 2 2 1 2 1 2 2 0 2
ϕ2,13 0 2 2 2 2 2 2 2 2 1 2 1 2 2 1 1
ϕ3,8 0 3 3 3 3 3 3 3 3 2 3 1 3 3 2 1
ϕ3,6 0 3 3 3 3 3 3 3 3 0 3 3 3 3 1 2
ϕ3,4 0 3 3 3 3 3 3 3 3 0 3 3 3 3 2 1
ϕ3,2 0 3 3 3 3 3 3 3 3 1 3 2 3 3 1 2
ϕ4,5 0 4 4 4 4 4 4 4 4 1 4 3 4 4 2 2
ϕ4,3 0 4 4 4 4 4 4 4 4 1 4 3 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 t7 t7 t10 0 t8 0 t4 t2 0 t3
ϕ1,6 0 1 t6 t12 t7 t10 t13 t t4 t7 t2 0 t6 t4 t3 0
ϕ1,12 0 t18 1 t6 t13 t4 t7 t7 t10 t t4 0 t8 t6 0 t3
ϕ1,18 0 t12 t18 1 t7 t10 t t13 t4 0 t6 t4 t2 t8 t3 0
ϕ2,1 0 t + t5 t7 + t11 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 0 t3 + t7 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 0 t10 + t14 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 t6 + t14 0 t4 + t8 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t t7 + t11 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 0 t5 + t9 t3 t + t5 t3 + t7 t2 t4
ϕ2,7'' 0 t7 + t11 t + t5 t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 0 t5 + t9 t3 + t7 t2 t4
ϕ2,10 0 t4 + t8 t10 + t14 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 t2 + t6 t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 0 t13 + t17 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 t3 + t7 t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 0 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 1 + t4 + t8 t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 0 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 0 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 0 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 0 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 0 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 t2 + 2t6 t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 0 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 t4 2t3 + 2t7 t + t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 t + 2t5 + t9 t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 + k1,2 − 3k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,6,  ϕ1,6,  ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,6611 + 2t + 3t2
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,6633 + 2t + t2
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54244 + 8t + 8t2 + 7t3 + 6t4 + 5t5 + 4t6
ϕ4,34244 + 5t + 6t2 + 7t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1
ϕ1,6 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0
ϕ1,12 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0
ϕ1,18 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0
ϕ2,1 2 1 2 2 2 2 2 2 2 2 2 0 2 2 0 1
ϕ2,4 2 0 2 2 2 2 2 2 2 2 2 0 2 2 2 0
ϕ2,7' 2 0 2 2 2 2 2 2 2 2 2 0 2 2 1 1
ϕ2,7'' 2 0 2 2 2 2 2 2 2 2 2 0 2 2 1 1
ϕ2,10 2 0 2 2 2 2 2 2 2 2 2 0 2 2 0 2
ϕ2,13 2 0 2 2 2 2 2 2 2 2 2 1 2 2 1 0
ϕ3,8 3 0 3 3 3 3 3 3 3 3 3 0 3 3 2 1
ϕ3,6 3 0 3 3 3 3 3 3 3 3 3 1 3 3 1 1
ϕ3,4 3 1 3 3 3 3 3 3 3 3 3 0 3 3 1 1
ϕ3,2 3 0 3 3 3 3 3 3 3 3 3 0 3 3 1 2
ϕ4,5 4 0 4 4 4 4 4 4 4 4 4 0 4 4 2 2
ϕ4,3 4 0 4 4 4 4 4 4 4 4 4 0 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 0 t12 t18 t t4 t7 t7 t10 t13 t8 0 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 t10 t13 t t4 t7 t2 0 t6 t4 0 0
ϕ1,12 t12 0 1 t6 t13 t4 t7 t7 t10 t t4 t2 t8 t6 0 0
ϕ1,18 t6 0 t18 1 t7 t10 t t13 t4 t7 t6 0 t2 t8 t3 0
ϕ2,1 t7 + t11 t t7 + t11 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 0 t3 + t7 t + t5 0 t2
ϕ2,4 t4 + t8 0 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 0 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 0 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 0 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 0 t + t5 t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 0 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 0 t10 + t14 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 0 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 0 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t t3 + t7 t5 + t9 t4 0
ϕ3,8 t8 + t12 + t16 0 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 0 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 0 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 t2 + 2t6 2t4 + t8 t3 t5
ϕ3,4 t4 + t8 + t12 t2 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 0 1 + t4 + t8 t2 + 2t6 t t3
ϕ3,2 t6 + t10 + t14 0 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 0 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 0 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 0 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 0 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 + k1,2 − 2k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,8,  ϕ1,6,  ϕ2,13},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,62711 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 4t6 + 2t7
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,13322 + t
ϕ3,86633 + 6t + 8t2 + 8t3 + 8t4 + 8t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1
ϕ1,6 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0
ϕ1,12 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1
ϕ1,18 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0
ϕ2,1 2 2 2 2 2 2 2 2 2 0 0 2 2 2 1 1
ϕ2,4 2 0 2 2 2 2 2 2 2 0 2 2 2 2 2 0
ϕ2,7' 2 1 2 2 2 2 2 2 2 0 1 2 2 2 1 1
ϕ2,7'' 2 0 2 2 2 2 2 2 2 0 2 2 2 2 1 1
ϕ2,10 2 1 2 2 2 2 2 2 2 0 1 2 2 2 0 2
ϕ2,13 2 0 2 2 2 2 2 2 2 1 1 2 2 2 1 1
ϕ3,8 3 0 3 3 3 3 3 3 3 0 3 3 3 3 2 1
ϕ3,6 3 0 3 3 3 3 3 3 3 0 3 3 3 3 1 2
ϕ3,4 3 2 3 3 3 3 3 3 3 0 1 3 3 3 2 1
ϕ3,2 3 1 3 3 3 3 3 3 3 0 2 3 3 3 1 2
ϕ4,5 4 1 4 4 4 4 4 4 4 0 3 4 4 4 2 2
ϕ4,3 4 1 4 4 4 4 4 4 4 0 3 4 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 t7 t7 t10 0 0 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 t12 t7 t10 t13 t t4 0 0 t8 t6 t4 t3 0
ϕ1,12 t12 0 1 t6 t13 t4 t7 t7 t10 t 0 t2 t8 t6 0 t3
ϕ1,18 t6 0 t18 1 t7 t10 t t13 t4 0 t6 t4 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 0 0 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 0 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 t6 + t14 0 t4 + t8 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 0 t5 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 0 t + t5 t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 + t11 0 t + t5 t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 t10 + t14 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 + t8 0 t2 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 0 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 t7 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 0 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 0 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 0 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 0 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 0 t4 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 0 t2 + t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 0 2t3 + t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 0 t + t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 + k1,2 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,5,  ϕ2,1,  ϕ4,3,  ϕ2,13}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,1822 + 4t + 2t2
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,13822 + 4t + 2t2
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,31684 + 8t + 4t2

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0
ϕ1,6 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0
ϕ1,12 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0
ϕ1,18 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0
ϕ2,1 2 2 2 2 1 2 2 2 2 0 2 2 2 2 1 0
ϕ2,4 2 2 2 2 0 2 2 2 2 0 2 2 2 2 2 0
ϕ2,7' 2 2 2 2 1 2 2 2 2 0 2 2 2 2 1 0
ϕ2,7'' 2 2 2 2 0 2 2 2 2 1 2 2 2 2 1 0
ϕ2,10 2 2 2 2 0 2 2 2 2 0 2 2 2 2 0 1
ϕ2,13 2 2 2 2 0 2 2 2 2 1 2 2 2 2 1 0
ϕ3,8 3 3 3 3 0 3 3 3 3 1 3 3 3 3 2 0
ϕ3,6 3 3 3 3 0 3 3 3 3 0 3 3 3 3 1 1
ϕ3,4 3 3 3 3 1 3 3 3 3 0 3 3 3 3 2 0
ϕ3,2 3 3 3 3 0 3 3 3 3 0 3 3 3 3 1 1
ϕ4,5 4 4 4 4 0 4 4 4 4 0 4 4 4 4 2 1
ϕ4,3 4 4 4 4 0 4 4 4 4 0 4 4 4 4 2 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 t7 t7 t10 0 t8 t6 t4 t2 0 0
ϕ1,6 t18 1 t6 t12 0 t10 t13 t t4 0 t2 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 t6 0 t4 t7 t7 t10 t t4 t2 t8 t6 0 0
ϕ1,18 t6 t12 t18 1 0 t10 t t13 t4 0 t6 t4 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 t13 + t17 1 t3 + t11 t6 + t14 t2 + t6 t5 + t9 0 t3 + t7 t5 + t9 t3 + t7 t + t5 t4 0
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 t10 + t14 0 1 + t8 t3 + t11 t3 + t11 t6 + t14 0 t4 + t8 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 t7 + t11 t2 t5 + t9 1 + t8 t8 + t12 t3 + t11 0 t5 + t9 t3 + t7 t + t5 t3 + t7 t2 0
ϕ2,7'' t13 + t17 t7 + t11 t + t5 t7 + t11 0 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 t + t5 t3 + t7 t5 + t9 t3 + t7 t2 0
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 t4 + t8 0 t6 + t14 t5 + t9 t5 + t9 1 + t8 0 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 t
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t + t5 0 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 t3 + t7 t + t5 t3 + t7 t5 + t9 t4 0
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 0 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 0
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 0 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 0 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 0 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 0
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 0 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 0 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 0 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 0 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 0 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 0 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1

For the generic point of the hyperplane k1,1 + k1,2 + k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,8,  ϕ1,0,  ϕ4,5,  ϕ4,3}

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
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,189611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,8633 + 2t + t2
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54244 + 5t + 6t2 + 7t3 + 8t4 + 8t5 + 4t6
ϕ4,34244 + 8t + 8t2 + 7t3 + 6t4 + 5t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0
ϕ1,6 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
ϕ1,12 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1
ϕ1,18 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0
ϕ2,1 0 2 2 2 2 2 2 2 2 2 0 2 2 2 1 1
ϕ2,4 0 2 2 2 2 2 2 2 2 2 0 2 2 2 2 0
ϕ2,7' 1 2 2 2 2 2 2 2 2 2 0 2 2 2 1 0
ϕ2,7'' 0 2 2 2 2 2 2 2 2 2 1 2 2 2 0 1
ϕ2,10 0 2 2 2 2 2 2 2 2 2 0 2 2 2 0 2
ϕ2,13 0 2 2 2 2 2 2 2 2 2 0 2 2 2 1 1
ϕ3,8 0 3 3 3 3 3 3 3 3 3 1 3 3 3 1 1
ϕ3,6 1 3 3 3 3 3 3 3 3 3 0 3 3 3 1 1
ϕ3,4 0 3 3 3 3 3 3 3 3 3 0 3 3 3 2 1
ϕ3,2 0 3 3 3 3 3 3 3 3 3 0 3 3 3 1 2
ϕ4,5 0 4 4 4 4 4 4 4 4 4 0 4 4 4 2 2
ϕ4,3 0 4 4 4 4 4 4 4 4 4 0 4 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 t18 t t4 t7 t7 t10 t13 0 t6 t4 t2 0 0
ϕ1,6 0 1 t6 t12 t7 t10 t13 t t4 t7 t2 t8 t6 t4 0 0
ϕ1,12 0 t18 1 t6 t13 t4 t7 t7 t10 t 0 t2 t8 t6 0 t3
ϕ1,18 0 t12 t18 1 t7 t10 t t13 t4 t7 0 t4 t2 t8 t3 0
ϕ2,1 0 t + t5 t7 + t11 t13 + t17 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 0 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 0 t10 + t14 t4 + t8 t10 + t14 t5 + t9 1 + t8 t3 + t11 t3 + t11 t6 + t14 t5 + t9 0 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t t7 + t11 t13 + t17 t7 + t11 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 t6 + t14 0 t3 + t7 t + t5 t3 + t7 t2 0
ϕ2,7'' 0 t7 + t11 t + t5 t7 + t11 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t t3 + t7 t5 + t9 t3 + t7 0 t4
ϕ2,10 0 t4 + t8 t10 + t14 t4 + t8 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 + t11 0 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 0 t13 + t17 t7 + t11 t + t5 t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 0 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 0 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t5 t3
ϕ3,6 t2 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 0 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t
ϕ3,4 0 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 0 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 0 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 0 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 0 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 0 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 0 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 0 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane 2k1,1 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,4,  ϕ1,18,  ϕ2,1},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,182711 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 4t6 + 2t7
ϕ2,1322 + t
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,46633 + 6t + 8t2 + 8t3 + 8t4 + 8t5 + 7t6 + 6t7 + 5t8 + 4t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1
ϕ1,6 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0
ϕ1,12 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1
ϕ1,18 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0
ϕ2,1 2 2 2 0 1 2 2 2 2 2 2 2 1 2 1 1
ϕ2,4 2 2 2 0 0 2 2 2 2 2 2 2 2 2 2 0
ϕ2,7' 2 2 2 0 0 2 2 2 2 2 2 2 2 2 1 1
ϕ2,7'' 2 2 2 1 0 2 2 2 2 2 2 2 1 2 1 1
ϕ2,10 2 2 2 1 0 2 2 2 2 2 2 2 1 2 0 2
ϕ2,13 2 2 2 2 0 2 2 2 2 2 2 2 0 2 1 1
ϕ3,8 3 3 3 2 0 3 3 3 3 3 3 3 1 3 2 1
ϕ3,6 3 3 3 1 0 3 3 3 3 3 3 3 2 3 1 2
ϕ3,4 3 3 3 0 0 3 3 3 3 3 3 3 3 3 2 1
ϕ3,2 3 3 3 0 0 3 3 3 3 3 3 3 3 3 1 2
ϕ4,5 4 4 4 1 0 4 4 4 4 4 4 4 3 4 2 2
ϕ4,3 4 4 4 1 0 4 4 4 4 4 4 4 3 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 0 t t4 t7 t7 t10 t13 t8 t6 0 t2 0 t3
ϕ1,6 t18 1 t6 0 0 t10 t13 t t4 t7 t2 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 t6 0 t4 t7 t7 t10 t t4 t2 0 t6 0 t3
ϕ1,18 t6 t12 t18 1 0 t10 t t13 t4 t7 t6 t4 0 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 0 1 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t7 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 0 0 1 + t8 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 0 0 t5 + t9 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 t7 0 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 t4 0 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t + t5 0 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 0 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 0 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 t4 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 t4 0 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 0 0 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 0 0 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 0 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 0 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,6,  ϕ1,18,  ϕ2,4},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,181211 + 2t + 3t2 + 4t3 + 2t4
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,41222 + 4t + 3t2 + 2t3 + t4
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,67233 + 6t + 7t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 7t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1
ϕ1,6 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0
ϕ1,12 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1
ϕ1,18 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0
ϕ2,1 2 2 2 0 2 1 2 2 2 2 2 1 2 2 1 1
ϕ2,4 2 2 2 0 2 1 2 2 2 2 2 1 2 2 2 0
ϕ2,7' 2 2 2 0 2 0 2 2 2 2 2 2 2 2 1 1
ϕ2,7'' 2 2 2 0 2 0 2 2 2 2 2 2 2 2 1 1
ϕ2,10 2 2 2 1 2 0 2 2 2 2 2 1 2 2 0 2
ϕ2,13 2 2 2 1 2 0 2 2 2 2 2 1 2 2 1 1
ϕ3,8 3 3 3 1 3 0 3 3 3 3 3 2 3 3 2 1
ϕ3,6 3 3 3 0 3 0 3 3 3 3 3 3 3 3 1 2
ϕ3,4 3 3 3 0 3 0 3 3 3 3 3 3 3 3 2 1
ϕ3,2 3 3 3 0 3 1 3 3 3 3 3 2 3 3 1 2
ϕ4,5 4 4 4 0 4 0 4 4 4 4 4 4 4 4 2 2
ϕ4,3 4 4 4 1 4 1 4 4 4 4 4 2 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 0 t t4 t7 t7 t10 t13 t8 0 t4 t2 0 t3
ϕ1,6 t18 1 t6 0 t7 0 t13 t t4 t7 t2 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 0 t13 0 t7 t7 t10 t t4 t2 t8 t6 0 t3
ϕ1,18 t6 t12 t18 1 t7 0 t t13 t4 t7 t6 0 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 0 1 + t8 t3 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 0 t5 + t9 1 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 0 t2 + t6 0 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 0 t6 + t14 0 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 t4 t3 + t11 0 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t t8 + t12 0 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t2 t5 + t9 + t13 0 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 0 t3 + t7 + t11 0 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 0 t + t5 + t9 0 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 0 t3 + t7 + t11 t2 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 0 2t4 + t8 + t12 0 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 t2 + t6 + 2t10 t 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 t3 + t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane 2k1,1 − k1,2 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,8,  ϕ1,18,  ϕ2,7''},   {ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,18311 + 2t
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''2722 + 4t + 6t2 + 5t3 + 4t4 + 3t5 + 2t6 + t7
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,86633 + 4t + 5t2 + 6t3 + 7t4 + 8t5 + 8t6 + 8t7 + 8t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,29633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ4,54844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6
ϕ4,34844 + 8t + 8t2 + 8t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1
ϕ1,6 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0
ϕ1,12 1 1 1 0 1 1 1 0 1 1 1 1 1 1 0 1
ϕ1,18 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0
ϕ2,1 2 2 2 0 2 2 2 2 2 2 0 2 2 2 1 1
ϕ2,4 2 2 2 0 2 2 2 1 2 2 1 2 2 2 2 0
ϕ2,7' 2 2 2 0 2 2 2 0 2 2 2 2 2 2 1 1
ϕ2,7'' 2 2 2 0 2 2 2 1 2 2 1 2 2 2 1 1
ϕ2,10 2 2 2 0 2 2 2 0 2 2 2 2 2 2 0 2
ϕ2,13 2 2 2 1 2 2 2 0 2 2 1 2 2 2 1 1
ϕ3,8 3 3 3 0 3 3 3 0 3 3 3 3 3 3 2 1
ϕ3,6 3 3 3 0 3 3 3 0 3 3 3 3 3 3 1 2
ϕ3,4 3 3 3 0 3 3 3 1 3 3 2 3 3 3 2 1
ϕ3,2 3 3 3 0 3 3 3 2 3 3 1 3 3 3 1 2
ϕ4,5 4 4 4 0 4 4 4 1 4 4 3 4 4 4 2 2
ϕ4,3 4 4 4 0 4 4 4 1 4 4 3 4 4 4 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 0 t t4 t7 t7 t10 t13 0 t6 t4 t2 0 t3
ϕ1,6 t18 1 t6 0 t7 t10 t13 t t4 t7 0 t8 t6 t4 t3 0
ϕ1,12 t12 t18 1 0 t13 t4 t7 0 t10 t t4 t2 t8 t6 0 t3
ϕ1,18 t6 t12 t18 1 t7 t10 t 0 t4 t7 0 t4 t2 t8 t3 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 0 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 0 t5 + t9 t3 + t7 t + t5 t4 t2
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 0 t5 + t9 1 + t8 t3 + t11 t3 t6 + t14 t5 + t9 t8 t2 + t6 t4 + t8 t2 + t6 t + t5 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 0 t2 + t6 t5 + t9 1 + t8 0 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 t3 + t7 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 0 t6 + t14 t5 + t9 t8 + t12 1 t3 + t11 t2 + t6 t5 t3 + t7 t5 + t9 t3 + t7 t2 t4
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 0 t3 + t11 t6 + t14 t5 + t9 0 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 t4 + t8 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t t8 + t12 t3 + t11 t2 + t6 0 t5 + t9 1 + t8 t3 t + t5 t3 + t7 t5 + t9 t4 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 0 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 0 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 t2 + t6 + t10 t + t5 t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 0 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 0 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 2t4 + t8 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 0 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 t2 + t6 + t10 t5 + t9 + t13 t4 + t8 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 0 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 t4 + t8 + t12 t3 + t7 + t11 t6 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 t3 t + t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 0 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 t + t5 + t9 + t13 2t4 + t8 + t12 t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 0 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 t4 t3 + 2t7 + t11 t2 + t6 + 2t10 t + t5 + t9 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 t2 + t6 1 + t4

For the generic point of the hyperplane 3k1,1 − k1,2 − k1,3

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,2,  ϕ1,18,  ϕ4,5,  ϕ4,3}

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 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,69611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,129611 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 7t12 + 6t13 + 5t14 + 4t15 + 3t16 + 2t17 + t18
ϕ1,18611 + 2t + 3t2
ϕ2,19622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,49622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7'9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,7''9622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,109622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ2,139622 + 4t + 6t2 + 8t3 + 8t4 + 8t5 + 8t6 + 8t7 + 8t8 + 8t9 + 8t10 + 8t11 + 6t12 + 4t13 + 2t14
ϕ3,89633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,69633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,49633 + 6t + 9t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 9t8 + 6t9 + 3t10
ϕ3,2633 + 2t + t2
ϕ4,54244 + 8t + 8t2 + 7t3 + 6t4 + 5t5 + 4t6
ϕ4,34244 + 5t + 6t2 + 7t3 + 8t4 + 8t5 + 4t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0
ϕ1,6 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0
ϕ1,12 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1
ϕ1,18 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0
ϕ2,1 2 2 2 0 2 2 2 2 2 2 2 2 2 1 1 0
ϕ2,4 2 2 2 0 2 2 2 2 2 2 2 2 2 0 2 0
ϕ2,7' 2 2 2 0 2 2 2 2 2 2 2 2 2 0 1 1
ϕ2,7'' 2 2 2 0 2 2 2 2 2 2 2 2 2 0 1 1
ϕ2,10 2 2 2 0 2 2 2 2 2 2 2 2 2 0 0 2
ϕ2,13 2 2 2 1 2 2 2 2 2 2 2 2 2 0 0 1
ϕ3,8 3 3 3 1 3 3 3 3 3 3 3 3 3 0 1 1
ϕ3,6 3 3 3 0 3 3 3 3 3 3 3 3 3 0 1 2
ϕ3,4 3 3 3 0 3 3 3 3 3 3 3 3 3 0 2 1
ϕ3,2 3 3 3 0 3 3 3 3 3 3 3 3 3 1 1 1
ϕ4,5 4 4 4 0 4 4 4 4 4 4 4 4 4 0 2 2
ϕ4,3 4 4 4 0 4 4 4 4 4 4 4 4 4 0 2 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,6) L(ϕ1,12) L(ϕ1,18) L(ϕ2,1) L(ϕ2,4) L(ϕ2,7') L(ϕ2,7'') L(ϕ2,10) L(ϕ2,13) L(ϕ3,8) L(ϕ3,6) L(ϕ3,4) L(ϕ3,2) L(ϕ4,5) L(ϕ4,3)
ϕ1,0 1 t6 t12 0 t t4 t7 t7 t10 t13 t8 t6 t4 t2 0 0
ϕ1,6 t18 1 t6 0 t7 t10 t13 t t4 t7 t2 t8 t6 0 t3 0
ϕ1,12 t12 t18 1 0 t13 t4 t7 t7 t10 t t4 t2 t8 0 0 t3
ϕ1,18 t6 t12 t18 1 t7 t10 t t13 t4 t7 t6 t4 t2 0 0 0
ϕ2,1 t7 + t11 t + t5 t7 + t11 0 1 + t8 t3 + t11 t6 + t14 t2 + t6 t5 + t9 t8 + t12 t3 + t7 t5 + t9 t3 + t7 t t4 0
ϕ2,4 t4 + t8 t10 + t14 t4 + t8 0 t5 + t9 1 + t8 t3 + t11 t3 + t11 t6 + t14 t5 + t9 t4 + t8 t2 + t6 t4 + t8 0 t + t5 0
ϕ2,7' t + t5 t7 + t11 t13 + t17 0 t2 + t6 t5 + t9 1 + t8 t8 + t12 t3 + t11 t6 + t14 t5 + t9 t3 + t7 t + t5 0 t2 t4
ϕ2,7'' t13 + t17 t7 + t11 t + t5 0 t6 + t14 t5 + t9 t8 + t12 1 + t8 t3 + t11 t2 + t6 t + t5 t3 + t7 t5 + t9 0 t2 t4
ϕ2,10 t10 + t14 t4 + t8 t10 + t14 0 t3 + t11 t6 + t14 t5 + t9 t5 + t9 1 + t8 t3 + t11 t2 + t6 t4 + t8 t2 + t6 0 0 t + t5
ϕ2,13 t7 + t11 t13 + t17 t7 + t11 t t8 + t12 t3 + t11 t2 + t6 t6 + t14 t5 + t9 1 + t8 t3 + t7 t + t5 t3 + t7 0 0 t2
ϕ3,8 t8 + t12 + t16 t6 + t10 + t14 t4 + t8 + t12 t2 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 1 + t4 + t8 t2 + 2t6 2t4 + t8 0 t t3
ϕ3,6 t2 + t6 + t10 t8 + t12 + t16 t6 + t10 + t14 0 t3 + t7 + t11 t2 + t6 + t10 t + t5 + t9 t5 + t9 + t13 t4 + t8 + t12 t3 + t7 + t11 t2 + t6 + t10 1 + t4 + t8 t2 + 2t6 0 t3 t + t5
ϕ3,4 t4 + t8 + t12 t2 + t6 + t10 t8 + t12 + t16 0 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 2t4 + t8 t2 + t6 + t10 1 + t4 + t8 0 t + t5 t3
ϕ3,2 t6 + t10 + t14 t4 + t8 + t12 t2 + t6 + t10 0 t3 + t7 + t11 t2 + t6 + t10 t5 + t9 + t13 t + t5 + t9 t4 + t8 + t12 t3 + t7 + t11 t2 + 2t6 2t4 + t8 t2 + t6 + t10 1 t3 t5
ϕ4,5 t3 + t7 + t11 + t15 t5 + 2t9 + t13 t3 + t7 + t11 + t15 0 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t3 + 2t7 t + 2t5 + t9 2t3 + 2t7 0 1 + t4 t2 + t6
ϕ4,3 t5 + 2t9 + t13 t3 + t7 + t11 + t15 t5 + 2t9 + t13 0 t2 + t6 + 2t10 t + t5 + t9 + t13 2t4 + t8 + t12 2t4 + t8 + t12 t3 + 2t7 + t11 t2 + t6 + 2t10 t + 2t5 + t9 2t3 + 2t7 t + 2t5 + t9 0 t2 + t6 1 + t4