The representation theory of the restricted rational Cherednik algebra for G20

Computed by Ulrich Thiel using CHAMP (see LMS J. Comput. Math., 2015). Last update on Fri Mar 27 12:48:20 CET 2015.

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

Quick navigation: Exceptional hyperplanes

For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,11,  ϕ2,17},   {ϕ2,21,  ϕ2,27},   {ϕ4,3,  ϕ4,6},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2},   {ϕ4,13,  ϕ4,16},   {ϕ2,1,  ϕ2,7},   {ϕ4,11,  ϕ4,8}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1118044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,818044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 0 0
ϕ1,20 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0
ϕ1,40 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0
ϕ2,21 2 2 2 2 0 2 0 2 0 0 0 1 0 1 0 1 2 1 1 0 1 2 2 2 0 0 0
ϕ2,27 2 2 2 0 1 2 0 0 1 0 0 0 0 0 0 1 0 1 1 2 1 2 2 2 0 0 1
ϕ2,11 2 2 2 2 0 2 0 2 0 1 0 0 0 1 0 1 1 2 1 1 0 2 2 2 0 0 0
ϕ2,17 2 2 2 0 1 0 1 2 0 0 0 0 0 0 0 1 1 0 1 1 2 2 2 2 1 0 0
ϕ2,1 2 2 2 2 0 2 0 2 0 1 0 1 0 0 0 2 1 1 0 1 1 2 2 2 0 0 0
ϕ2,7 2 2 2 2 0 0 1 0 1 0 0 0 0 0 0 0 1 1 2 1 1 2 2 2 0 1 0
ϕ3,2 3 3 3 3 0 3 0 1 1 1 0 1 0 1 0 2 2 2 1 1 1 3 3 3 0 0 0
ϕ3,14 3 3 3 1 1 1 1 3 0 0 1 0 0 0 0 2 2 0 1 1 3 3 3 3 0 0 0
ϕ3,10'' 3 3 3 1 1 3 0 3 0 1 0 1 0 1 0 2 2 2 1 1 1 3 3 3 0 0 0
ϕ3,10' 3 3 3 3 0 1 1 1 1 0 0 0 1 0 0 0 2 2 3 1 1 3 3 3 0 0 0
ϕ3,12 3 3 3 3 0 1 1 3 0 1 0 1 0 1 0 2 2 2 1 1 1 3 3 3 0 0 0
ϕ3,6 3 3 3 1 1 3 0 1 1 0 0 0 0 0 1 2 0 2 1 3 1 3 3 3 0 0 0
ϕ4,3 4 4 4 2 1 4 0 2 1 0 0 1 0 1 0 3 2 3 1 2 1 4 4 4 0 0 1
ϕ4,11 4 4 4 2 1 2 1 4 0 1 0 0 0 1 0 3 3 2 1 1 2 4 4 4 1 0 0
ϕ4,13 4 4 4 4 0 2 1 2 1 1 0 1 0 0 0 2 3 3 2 1 1 4 4 4 0 1 0
ϕ4,6 4 4 4 2 1 2 1 2 1 0 0 0 0 0 0 1 2 1 3 2 3 4 4 4 1 1 0
ϕ4,8 4 4 4 2 1 2 1 2 1 0 0 0 0 0 0 1 1 2 3 3 2 4 4 4 0 1 1
ϕ4,16 4 4 4 2 1 2 1 2 1 0 0 0 0 0 0 2 1 1 2 3 3 4 4 4 1 0 1
ϕ5,12 5 5 5 3 1 3 1 3 1 1 0 0 0 0 0 3 2 3 2 3 2 5 5 5 1 1 0
ϕ5,8 5 5 5 3 1 3 1 3 1 0 0 1 0 0 0 3 3 2 2 2 3 5 5 5 0 1 1
ϕ5,4 5 5 5 3 1 3 1 3 1 0 0 0 0 1 0 2 3 3 3 2 2 5 5 5 1 0 1
ϕ6,7 6 6 6 2 2 4 1 4 1 0 0 0 0 0 0 4 2 2 2 4 4 6 6 6 1 1 1
ϕ6,9 6 6 6 4 1 2 2 4 1 0 0 0 0 0 0 2 4 2 4 2 4 6 6 6 1 1 1
ϕ6,5 6 6 6 4 1 4 1 2 2 0 0 0 0 0 0 2 2 4 4 4 2 6 6 6 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 t40 t21 0 t11 0 t 0 t2 0 0 0 0 0 t3 0 t13 0 t8 0 t12 t8 t4 0 0 0
ϕ1,20 t40 1 t20 t 0 t21 0 t11 0 0 0 t2 0 0 0 t13 t3 0 0 0 t8 t4 t12 t8 0 0 0
ϕ1,40 t20 t40 1 t11 0 t 0 t21 0 0 0 0 0 t2 0 0 t13 t3 t8 0 0 t8 t4 t12 0 0 0
ϕ2,21 t21 + t39 t11 + t29 t + t19 1 + t12 0 t2 + t20 0 t10 + t22 0 0 0 t 0 t3 0 t12 t2 + t14 t4 t9 0 t7 t3 + t9 t5 + t11 t7 + t13 0 0 0
ϕ2,27 t27 + t33 t17 + t23 t7 + t13 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 t6 0 t10 t15 t5 + t11 t t3 + t9 t5 + t11 t7 + t13 0 0 t2
ϕ2,11 t + t19 t21 + t39 t11 + t29 t10 + t22 0 1 + t12 0 t2 + t20 0 t3 0 0 0 t 0 t4 t12 t2 + t14 t7 t9 0 t7 + t13 t3 + t9 t5 + t11 0 0 0
ϕ2,17 t7 + t13 t27 + t33 t17 + t23 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 t10 t6 0 t t15 t5 + t11 t7 + t13 t3 + t9 t5 + t11 t2 0 0
ϕ2,1 t11 + t29 t + t19 t21 + t39 t2 + t20 0 t10 + t22 0 1 + t12 0 t 0 t3 0 0 0 t2 + t14 t4 t12 0 t7 t9 t5 + t11 t7 + t13 t3 + t9 0 0 0
ϕ2,7 t17 + t23 t7 + t13 t27 + t33 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 t10 t6 t5 + t11 t t15 t5 + t11 t7 + t13 t3 + t9 0 t2 0
ϕ3,2 t10 + t22 + t28 t12 + t18 + t30 t2 + t20 + t38 t + t13 + t19 0 t3 + t9 + t21 0 t11 t5 1 0 t2 0 t4 0 t + t13 t3 + t15 t5 + t11 t10 t6 t8 t4 + 2t10 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,14 t10 + t16 + t34 t6 + t24 + t30 t14 + t20 + t26 t7 t7 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 t7 + t13 t3 + t9 0 t4 t12 t2 + t8 + t14 t4 + 2t10 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,10'' t2 + t20 + t38 t10 + t22 + t28 t12 + t18 + t30 t11 t5 t + t13 + t19 0 t3 + t9 + t21 0 t4 0 1 0 t2 0 t5 + t11 t + t13 t3 + t15 t8 t10 t6 t2 + t8 + t14 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,10' t14 + t20 + t26 t10 + t16 + t34 t6 + t24 + t30 t5 + t11 + t17 0 t7 t7 t15 t3 0 0 0 1 0 0 0 t7 + t13 t3 + t9 t2 + t8 + t14 t4 t12 t2 + t8 + t14 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,12 t12 + t18 + t30 t2 + t20 + t38 t10 + t22 + t28 t3 + t9 + t21 0 t11 t5 t + t13 + t19 0 t2 0 t4 0 1 0 t3 + t15 t5 + t11 t + t13 t6 t8 t10 2t6 + t12 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ3,6 t6 + t24 + t30 t14 + t20 + t26 t10 + t16 + t34 t15 t3 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 t3 + t9 0 t7 + t13 t12 t2 + t8 + t14 t4 2t6 + t12 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 t12 + t18 t6 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 t 0 t3 0 1 + t6 + t12 t2 + t14 t4 + t10 + t16 t9 t5 + t11 t7 t3 + 2t9 + t15 2t5 + 2t11 t + 2t7 + t13 0 0 t2
ϕ4,11 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 t4 + t10 t4 t12 + t18 t6 t2 + t8 + t14 + t20 0 t3 0 0 0 t 0 t4 + t10 + t16 1 + t6 + t12 t2 + t14 t7 t9 t5 + t11 t + 2t7 + t13 t3 + 2t9 + t15 2t5 + 2t11 t2 0 0
ϕ4,13 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 t2 + t8 + t14 + t20 0 t4 + t10 t4 t12 + t18 t6 t 0 t3 0 0 0 t2 + t14 t4 + t10 + t16 1 + t6 + t12 t5 + t11 t7 t9 2t5 + 2t11 t + 2t7 + t13 t3 + 2t9 + t15 0 t2 0
ϕ4,6 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 t9 + t15 t9 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 t9 t5 + t11 t7 1 + t6 + t12 t2 + t14 t4 + t10 + t16 2t6 + 2t12 t2 + 2t8 + t14 2t4 + 2t10 t t3 0
ϕ4,8 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 t7 + t13 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 t7 t9 t5 + t11 t4 + t10 + t16 1 + t6 + t12 t2 + t14 2t4 + 2t10 2t6 + 2t12 t2 + 2t8 + t14 0 t t3
ϕ4,16 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 t5 + t17 t5 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 t5 + t11 t7 t9 t2 + t14 t4 + t10 + t16 1 + t6 + t12 t2 + 2t8 + t14 2t4 + 2t10 2t6 + 2t12 t3 0 t
ϕ5,12 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 t3 + t9 + t15 t3 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t2 0 0 0 0 0 t3 + t9 + t15 t5 + t11 t + t7 + t13 t6 + t12 t2 + t8 + t14 t4 + t10 1 + 2t6 + 2t12 t2 + 3t8 + t14 2t4 + 2t10 + t16 t t3 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 t7 + t13 + t19 t7 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 t2 0 0 0 t + t7 + t13 t3 + t9 + t15 t5 + t11 t4 + t10 t6 + t12 t2 + t8 + t14 2t4 + 2t10 + t16 1 + 2t6 + 2t12 t2 + 3t8 + t14 0 t t3
ϕ5,4 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 t5 + t11 t + t7 + t13 t3 + t9 + t15 t2 + t8 + t14 t4 + t10 t6 + t12 t2 + 3t8 + t14 2t4 + 2t10 + t16 1 + 2t6 + 2t12 t3 0 t
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 t6 + t12 t5 + t11 t + t7 + 2t13 2t3 + t9 + t15 3t5 + 3t11 t + 3t7 + 2t13 2t3 + 3t9 + t15 1 t2 t4
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 t4 + t10 2t3 + t9 + t15 t5 + t11 t + t7 + 2t13 2t3 + 3t9 + t15 3t5 + 3t11 t + 3t7 + 2t13 t4 1 t2
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 t4 + t10 t6 + t12 t2 + 2t8 + t14 t + t7 + 2t13 2t3 + t9 + t15 t5 + t11 t + 3t7 + 2t13 2t3 + 3t9 + t15 3t5 + 3t11 t2 t4 1

Exceptional hyperplanes

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

For the generic point of the hyperplane k1,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,21,  ϕ2,27,  ϕ2,11,  ϕ2,17,  ϕ4,13,  ϕ4,16},   {ϕ4,3,  ϕ4,11,  ϕ4,6,  ϕ4,8},   {ϕ5,12,  ϕ5,8},   {ϕ1,0,  ϕ1,20,  ϕ2,1,  ϕ2,7},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,012111 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 10t11 + 9t12 + 8t13 + 7t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ1,20111
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,21822 + 4t + 2t2
ϕ2,274242 + 4t + 6t2 + 8t3 + 10t4 + 8t5 + 4t6
ϕ2,119622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 10t8 + 8t9 + 6t10 + 4t11 + 2t12
ϕ2,17242
ϕ2,14742 + 3t + 4t2 + 5t3 + 6t4 + 7t5 + 6t6 + 5t7 + 4t8 + 3t9 + 2t10
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,312044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 8t10 + 4t11
ϕ4,116044 + 8t + 12t2 + 12t3 + 12t4 + 8t5 + 4t6
ϕ4,134284 + 8t + 10t2 + 8t3 + 6t4 + 4t5 + 2t6
ϕ4,612044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 8t10 + 4t11
ϕ4,86044 + 8t + 12t2 + 12t3 + 12t4 + 8t5 + 4t6
ϕ4,1628124 + 6t + 8t2 + 6t3 + 4t4
ϕ5,1212055 + 10t + 15t2 + 20t3 + 20t4 + 20t5 + 15t6 + 10t7 + 5t8
ϕ5,824055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 25t8 + 20t9 + 15t10 + 10t11 + 5t12
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0
ϕ1,20 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0
ϕ1,40 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0
ϕ2,21 0 0 2 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 2 0 0 0
ϕ2,27 0 0 2 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 2 0 0 1
ϕ2,11 2 0 2 0 0 2 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 2 2 0 0 0
ϕ2,17 2 0 2 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 2 2 1 0 0
ϕ2,1 0 0 2 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 2 0 0 0
ϕ2,7 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 2 0 1 0
ϕ3,2 1 0 3 1 0 2 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 2 3 0 0 0
ϕ3,14 1 0 3 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 2 3 0 0 0
ϕ3,10'' 1 0 3 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 2 3 0 0 0
ϕ3,10' 1 0 3 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 2 1 0 1 2 3 0 0 0
ϕ3,12 1 0 3 0 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 2 3 0 0 0
ϕ3,6 1 0 3 0 1 1 0 0 1 0 0 0 0 0 1 2 0 0 0 1 0 1 2 3 0 0 0
ϕ4,3 2 0 4 0 1 2 0 0 1 0 0 1 0 1 0 2 1 0 1 0 0 1 3 4 0 0 1
ϕ4,11 0 0 4 0 1 0 0 2 0 1 0 0 0 1 0 1 2 1 1 0 0 2 2 4 1 0 0
ϕ4,13 2 0 4 0 0 2 0 0 1 1 0 1 0 0 0 1 1 1 2 0 0 1 3 4 0 1 0
ϕ4,6 2 0 4 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 2 1 1 1 3 4 1 1 0
ϕ4,8 0 0 4 0 1 0 0 1 1 0 0 0 0 0 0 1 0 1 1 2 0 2 2 4 0 1 1
ϕ4,16 2 0 4 0 0 1 0 0 1 0 0 0 0 0 0 2 0 0 1 1 1 1 3 4 1 0 1
ϕ5,12 1 0 5 0 1 1 0 1 1 1 0 0 0 0 0 2 1 1 1 1 0 2 3 5 1 1 0
ϕ5,8 3 0 5 0 0 2 0 0 1 0 0 1 0 0 0 2 1 0 2 0 1 1 4 5 0 1 1
ϕ5,4 1 0 5 0 1 1 0 1 1 0 0 0 0 1 0 1 1 1 2 1 0 2 3 5 1 0 1
ϕ6,7 2 0 6 0 1 1 0 1 1 0 0 0 0 0 0 3 1 0 1 1 1 2 4 6 1 1 1
ϕ6,9 2 0 6 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 3 1 1 2 4 6 1 1 1
ϕ6,5 2 0 6 0 1 2 0 0 2 0 0 0 0 0 0 2 0 1 2 2 0 2 4 6 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 t40 0 0 t11 0 0 0 t2 0 0 0 0 0 t3 0 0 0 0 0 0 t8 t4 0 0 0
ϕ1,20 0 1 t20 t 0 0 0 0 0 0 0 t2 0 0 0 0 t3 0 0 0 0 t4 0 t8 0 0 0
ϕ1,40 t20 0 1 0 0 t 0 0 0 0 0 0 0 t2 0 0 0 0 t8 0 0 0 t4 t12 0 0 0
ϕ2,21 0 0 t + t19 1 0 t2 0 t10 0 0 0 t 0 t3 0 0 t2 0 t9 0 0 t3 t5 t7 + t13 0 0 0
ϕ2,27 0 0 t7 + t13 0 1 0 0 0 t10 0 0 0 0 0 0 t6 0 0 0 t5 0 t3 t5 t7 + t13 0 0 t2
ϕ2,11 t + t19 0 t11 + t29 0 0 1 + t12 0 0 0 t3 0 0 0 t 0 t4 0 0 t7 0 0 0 t3 + t9 t5 + t11 0 0 0
ϕ2,17 t7 + t13 0 t17 + t23 0 0 0 1 0 0 0 0 0 0 0 0 t10 0 0 t 0 0 0 t3 + t9 t5 + t11 t2 0 0
ϕ2,1 0 0 t21 + t39 t2 0 t10 0 1 0 t 0 t3 0 0 0 t2 t4 0 0 0 0 t5 t7 t3 + t9 0 0 0
ϕ2,7 0 0 t27 + t33 0 0 0 0 0 1 0 0 0 0 0 0 0 0 t6 t5 t 0 t5 t7 t3 + t9 0 t2 0
ϕ3,2 t10 0 t2 + t20 + t38 t 0 t3 + t9 0 0 t5 1 0 t2 0 t4 0 t t3 0 t10 0 0 t4 2t6 t2 + t8 + t14 0 0 0
ϕ3,14 t10 0 t14 + t20 + t26 0 0 0 0 t5 0 0 1 0 0 0 0 t7 t3 0 t4 0 t2 t4 2t6 t2 + t8 + t14 0 0 0
ϕ3,10'' t2 0 t12 + t18 + t30 0 t5 t 0 t9 0 t4 0 1 0 t2 0 t5 t 0 t8 0 0 t2 t4 + t10 2t6 + t12 0 0 0
ϕ3,10' t14 0 t6 + t24 + t30 0 0 t7 0 0 t3 0 0 0 1 0 0 0 0 t3 t2 + t8 t4 0 t2 t4 + t10 2t6 + t12 0 0 0
ϕ3,12 t18 0 t10 + t22 + t28 0 0 t11 0 t 0 t2 0 t4 0 1 0 t3 t5 t t6 0 0 t6 t2 + t8 t4 + 2t10 0 0 0
ϕ3,6 t6 0 t10 + t16 + t34 0 t3 t5 0 0 t7 0 0 0 0 0 1 t3 + t9 0 0 0 t2 0 t6 t2 + t8 t4 + 2t10 0 0 0
ϕ4,3 t3 + t9 0 t13 + t19 + t31 + t37 0 t6 t2 + t8 0 0 t4 0 0 t 0 t3 0 1 + t6 t2 0 t9 0 0 t3 2t5 + t11 t + 2t7 + t13 0 0 t2
ϕ4,11 0 0 t11 + t17 + t23 + t29 0 t4 0 0 t2 + t8 0 t3 0 0 0 t 0 t4 1 + t6 t2 t7 0 0 t + t7 t3 + t9 2t5 + 2t11 t2 0 0
ϕ4,13 t11 + t17 0 t3 + t9 + t21 + t27 0 0 t4 + t10 0 0 t6 t 0 t3 0 0 0 t2 t4 1 t5 + t11 0 0 t5 t + 2t7 t3 + 2t9 + t15 0 t2 0
ϕ4,6 t6 + t12 0 t16 + t22 + t28 + t34 0 0 t5 0 0 t 0 0 0 0 0 0 t9 0 0 1 + t6 t2 t4 t6 t2 + 2t8 2t4 + 2t10 t t3 0
ϕ4,8 0 0 t8 + t14 + t26 + t32 0 t 0 0 t5 t5 0 0 0 0 0 0 t7 0 t5 t4 1 + t6 0 2t4 2t6 t2 + 2t8 + t14 0 t t3
ϕ4,16 t8 + t14 0 t6 + t12 + t18 + t24 0 0 t7 0 0 t9 0 0 0 0 0 0 t5 + t11 0 0 t2 t4 1 t2 2t4 + t10 2t6 + 2t12 t3 0 t
ϕ5,12 t12 0 t4 + t10 + t16 + t22 + t28 0 t3 t5 0 t7 t7 t2 0 0 0 0 0 t3 + t9 t5 t t6 t2 0 1 + t6 t2 + 2t8 2t4 + 2t10 + t16 t t3 0
ϕ5,8 t4 + t10 + t16 0 t8 + t14 + t20 + t26 + t32 0 0 t3 + t9 0 0 t5 0 0 t2 0 0 0 t + t7 t3 0 t4 + t10 0 t2 t4 1 + 2t6 + t12 t2 + 3t8 + t14 0 t t3
ϕ5,4 t8 0 t12 + t18 + t24 + t30 + t36 0 t5 t7 0 t3 t3 0 0 0 0 t2 0 t5 t t3 t2 + t8 t4 0 t2 + t8 2t4 + t10 1 + 2t6 + 2t12 t3 0 t
ϕ6,7 t5 + t11 0 t9 + 2t15 + t21 + t27 + t33 0 t2 t4 0 t6 t6 0 0 0 0 0 0 t2 + 2t8 t4 0 t5 t t3 2t5 t + 3t7 2t3 + 3t9 + t15 1 t2 t4
ϕ6,9 t9 + t15 0 t7 + t13 + t19 + 2t25 + t31 0 0 t8 0 t4 t4 0 0 0 0 0 0 t6 t2 t4 2t3 + t9 t5 t 2t3 3t5 + t11 t + 3t7 + 2t13 t4 1 t2
ϕ6,5 t7 + t13 0 t5 + t11 + t17 + t23 + t29 + t35 0 t4 2t6 0 0 t2 + t8 0 0 0 0 0 0 t4 + t10 0 t2 t + t7 2t3 0 t + t7 2t3 + 2t9 3t5 + 3t11 t2 t4 1

For the generic point of the hyperplane k1,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,3,  ϕ4,13,  ϕ4,6,  ϕ4,16},   {ϕ5,12,  ϕ5,4},   {ϕ2,21,  ϕ2,27,  ϕ4,11,  ϕ2,1,  ϕ2,7,  ϕ4,8},   {ϕ1,0,  ϕ1,40,  ϕ2,11,  ϕ2,17},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  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
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,4012111 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 10t11 + 9t12 + 8t13 + 7t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ2,219622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 10t8 + 8t9 + 6t10 + 4t11 + 2t12
ϕ2,27242
ϕ2,114742 + 3t + 4t2 + 5t3 + 6t4 + 7t5 + 6t6 + 5t7 + 4t8 + 3t9 + 2t10
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1822 + 4t + 2t2
ϕ2,74242 + 4t + 6t2 + 8t3 + 10t4 + 8t5 + 4t6
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,36044 + 8t + 12t2 + 12t3 + 12t4 + 8t5 + 4t6
ϕ4,114284 + 8t + 10t2 + 8t3 + 6t4 + 4t5 + 2t6
ϕ4,1312044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 8t10 + 4t11
ϕ4,66044 + 8t + 12t2 + 12t3 + 12t4 + 8t5 + 4t6
ϕ4,828124 + 6t + 8t2 + 6t3 + 4t4
ϕ4,1612044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 8t10 + 4t11
ϕ5,1224055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 25t8 + 20t9 + 15t10 + 10t11 + 5t12
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,412055 + 10t + 15t2 + 20t3 + 20t4 + 20t5 + 15t6 + 10t7 + 5t8
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0
ϕ1,20 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0
ϕ1,40 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0
ϕ2,21 0 2 2 2 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 2 2 0 0 0 0
ϕ2,27 0 2 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 2 2 0 0 0 1
ϕ2,11 0 2 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 2 1 0 0 0
ϕ2,17 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 2 1 1 0 0
ϕ2,1 0 2 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 2 1 0 0 0
ϕ2,7 0 2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 2 1 0 1 0
ϕ3,2 0 3 1 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 2 3 1 0 0 0
ϕ3,14 0 3 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 2 2 3 1 0 0 0
ϕ3,10'' 0 3 1 1 0 1 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 2 3 1 0 0 0
ϕ3,10' 0 3 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 2 1 0 0 2 3 1 0 0 0
ϕ3,12 0 3 1 2 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 2 3 1 0 0 0
ϕ3,6 0 3 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 2 3 1 0 0 0
ϕ4,3 0 4 0 0 0 2 0 0 1 0 0 1 0 1 0 2 1 1 0 0 1 2 4 2 0 0 1
ϕ4,11 0 4 2 2 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 2 3 4 1 1 0 0
ϕ4,13 0 4 2 2 0 0 1 0 1 1 0 1 0 0 0 1 0 2 0 0 1 3 4 1 0 1 0
ϕ4,6 0 4 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 2 0 1 2 4 2 1 1 0
ϕ4,8 0 4 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 1 1 3 4 1 0 1 1
ϕ4,16 0 4 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 2 3 4 1 1 0 1
ϕ5,12 0 5 3 2 0 0 1 0 0 1 0 0 0 0 0 1 0 2 0 1 2 4 5 1 1 1 0
ϕ5,8 0 5 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 1 1 0 2 3 5 2 0 1 1
ϕ5,4 0 5 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 2 1 0 1 3 5 2 1 0 1
ϕ6,7 0 6 2 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 3 4 6 2 1 1 1
ϕ6,9 0 6 2 2 0 0 2 0 1 0 0 0 0 0 0 0 1 2 2 0 2 4 6 2 1 1 1
ϕ6,5 0 6 2 1 0 1 1 0 1 0 0 0 0 0 0 1 0 3 1 1 1 4 6 2 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 0 0 0 0 0 t 0 t2 0 0 0 0 0 t3 0 0 0 0 0 0 t8 t4 0 0 0
ϕ1,20 0 1 t20 t 0 0 0 0 0 0 0 t2 0 0 0 0 0 0 0 0 t8 t4 t12 0 0 0 0
ϕ1,40 0 t40 1 t11 0 0 0 0 0 0 0 0 0 t2 0 0 0 t3 0 0 0 t8 t4 0 0 0 0
ϕ2,21 0 t11 + t29 t + t19 1 + t12 0 0 0 0 0 0 0 t 0 t3 0 0 0 t4 0 0 t7 t3 + t9 t5 + t11 0 0 0 0
ϕ2,27 0 t17 + t23 t7 + t13 0 1 0 0 0 0 0 0 0 0 0 0 0 0 t10 0 0 t t3 + t9 t5 + t11 0 0 0 t2
ϕ2,11 0 t21 + t39 0 t10 0 1 0 t2 0 t3 0 0 0 t 0 t4 0 t2 0 0 0 t7 t3 + t9 t5 0 0 0
ϕ2,17 0 t27 + t33 0 0 0 0 1 0 0 0 0 0 0 0 0 0 t6 0 t 0 t5 t7 t3 + t9 t5 t2 0 0
ϕ2,1 0 t + t19 0 t2 0 t10 0 1 0 t 0 t3 0 0 0 t2 0 0 0 0 t9 t5 t7 + t13 t3 0 0 0
ϕ2,7 0 t7 + t13 0 0 0 0 t10 0 1 0 0 0 0 0 0 0 0 t6 t5 0 0 t5 t7 + t13 t3 0 t2 0
ϕ3,2 0 t12 + t18 + t30 t2 t 0 t9 0 0 t5 1 0 t2 0 t4 0 t 0 t5 0 0 t8 t4 + t10 2t6 + t12 t2 0 0 0
ϕ3,14 0 t6 + t24 + t30 t14 t7 0 0 t3 0 0 0 1 0 0 0 0 0 t3 0 t4 0 t2 + t8 t4 + t10 2t6 + t12 t2 0 0 0
ϕ3,10'' 0 t10 + t22 + t28 t18 t11 0 t 0 0 0 t4 0 1 0 t2 0 t5 t t3 0 0 t6 t2 + t8 t4 + 2t10 t6 0 0 0
ϕ3,10' 0 t10 + t16 + t34 t6 t5 0 0 t7 0 t3 0 0 0 1 0 0 0 0 t3 + t9 t2 0 0 t2 + t8 t4 + 2t10 t6 0 0 0
ϕ3,12 0 t2 + t20 + t38 t10 t3 + t9 0 0 t5 t 0 t2 0 t4 0 1 0 t3 0 t 0 0 t10 2t6 t2 + t8 + t14 t4 0 0 0
ϕ3,6 0 t14 + t20 + t26 t10 0 0 t5 0 0 0 0 0 0 0 0 1 t3 0 t7 0 t2 t4 2t6 t2 + t8 + t14 t4 0 0 0
ϕ4,3 0 t11 + t17 + t23 + t29 0 0 0 t2 + t8 0 0 t4 0 0 t 0 t3 0 1 + t6 t2 t4 0 0 t7 t3 + t9 2t5 + 2t11 t + t7 0 0 t2
ϕ4,11 0 t3 + t9 + t21 + t27 t11 + t17 t4 + t10 0 0 t6 0 0 t3 0 0 0 t 0 t4 1 t2 0 0 t5 + t11 t + 2t7 t3 + 2t9 + t15 t5 t2 0 0
ϕ4,13 0 t13 + t19 + t31 + t37 t3 + t9 t2 + t8 0 0 t4 0 t6 t 0 t3 0 0 0 t2 0 1 + t6 0 0 t9 2t5 + t11 t + 2t7 + t13 t3 0 t2 0
ϕ4,6 0 t8 + t14 + t26 + t32 0 0 0 t5 t5 0 t 0 0 0 0 0 0 0 t5 t7 1 + t6 0 t4 2t6 t2 + 2t8 + t14 2t4 t t3 0
ϕ4,8 0 t6 + t12 + t18 + t24 t8 + t14 t7 0 0 t9 0 0 0 0 0 0 0 0 0 0 t5 + t11 t4 1 t2 2t4 + t10 2t6 + 2t12 t2 0 t t3
ϕ4,16 0 t16 + t22 + t28 + t34 t6 + t12 t5 0 0 t 0 0 0 0 0 0 0 0 0 0 t9 t2 t4 1 + t6 t2 + 2t8 2t4 + 2t10 t6 t3 0 t
ϕ5,12 0 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 t3 + t9 0 0 t5 0 0 t2 0 0 0 0 0 t3 0 t + t7 0 t2 t4 + t10 1 + 2t6 + t12 t2 + 3t8 + t14 t4 t t3 0
ϕ5,8 0 t12 + t18 + t24 + t30 + t36 t8 t7 0 t3 t3 0 t5 0 0 t2 0 0 0 t t3 t5 t4 0 t2 + t8 2t4 + t10 1 + 2t6 + 2t12 t2 + t8 0 t t3
ϕ5,4 0 t4 + t10 + t16 + t22 + t28 t12 t5 0 t7 t7 0 t3 0 0 0 0 t2 0 t5 t t3 + t9 t2 0 t6 t2 + 2t8 2t4 + 2t10 + t16 1 + t6 t3 0 t
ϕ6,7 0 t7 + t13 + t19 + 2t25 + t31 t9 + t15 t8 0 t4 t4 0 0 0 0 0 0 0 0 t2 t4 t6 t5 t 2t3 + t9 3t5 + t11 t + 3t7 + 2t13 2t3 1 t2 t4
ϕ6,9 0 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 2t6 0 0 t2 + t8 0 t4 0 0 0 0 0 0 0 t2 t4 + t10 2t3 0 t + t7 2t3 + 2t9 3t5 + 3t11 t + t7 t4 1 t2
ϕ6,5 0 t9 + 2t15 + t21 + t27 + t33 t5 + t11 t4 0 t6 t6 0 t2 0 0 0 0 0 0 t4 0 t2 + 2t8 t t3 t5 t + 3t7 2t3 + 3t9 + t15 2t5 t2 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,11,  ϕ2,17},   {ϕ1,20,  ϕ5,4,  ϕ4,3,  ϕ4,6},   {ϕ2,21,  ϕ2,27},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2},   {ϕ4,13,  ϕ4,16},   {ϕ2,1,  ϕ2,7},   {ϕ4,11,  ϕ4,8}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,201011 + 2t + 3t2 + 4t3
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,317044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 11t13 + 10t14 + 5t15
ϕ4,1118044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,64044 + 8t + 7t2 + 6t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ4,818044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,414055 + 6t + 7t2 + 8t3 + 9t4 + 10t5 + 11t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 8t13 + 4t14
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0
ϕ1,20 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0
ϕ1,40 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 0
ϕ2,21 2 0 2 2 0 2 0 2 0 0 0 1 0 1 0 1 2 1 0 0 1 2 2 1 0 0 0
ϕ2,27 2 0 2 0 1 2 0 0 1 0 0 0 0 0 0 1 0 1 0 2 1 2 2 1 0 0 1
ϕ2,11 2 0 2 2 0 2 0 2 0 1 0 0 0 1 0 1 1 2 1 1 0 2 2 0 0 0 0
ϕ2,17 2 0 2 0 1 0 1 2 0 0 0 0 0 0 0 1 1 0 1 1 2 2 2 0 1 0 0
ϕ2,1 2 1 2 2 0 2 0 2 0 1 0 1 0 0 0 1 1 1 0 1 1 2 2 0 0 0 0
ϕ2,7 2 0 2 2 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 2 2 2 0 1 0
ϕ3,2 3 0 3 3 0 3 0 1 1 1 0 1 0 1 0 2 2 2 0 1 1 3 3 1 0 0 0
ϕ3,14 3 0 3 1 1 1 1 3 0 0 1 0 0 0 0 2 2 0 0 1 3 3 3 1 0 0 0
ϕ3,10'' 3 0 3 1 1 3 0 3 0 1 0 1 0 1 0 2 2 2 0 1 1 3 3 1 0 0 0
ϕ3,10' 3 0 3 3 0 1 1 1 1 0 0 0 1 0 0 0 2 2 1 1 1 3 3 2 0 0 0
ϕ3,12 3 1 3 3 0 1 1 3 0 1 0 1 0 1 0 1 2 2 1 1 1 3 3 0 0 0 0
ϕ3,6 3 0 3 1 1 3 0 1 1 0 0 0 0 0 1 2 0 2 0 3 1 3 3 1 0 0 0
ϕ4,3 4 0 4 2 1 4 0 2 1 0 0 1 0 1 0 3 2 3 0 2 1 4 4 1 0 0 1
ϕ4,11 4 1 4 2 1 2 1 4 0 1 0 0 0 1 0 2 3 2 0 1 2 4 4 1 1 0 0
ϕ4,13 4 0 4 4 0 2 1 2 1 1 0 1 0 0 0 2 3 3 1 1 1 4 4 1 0 1 0
ϕ4,6 4 0 4 2 1 2 1 2 1 0 0 0 0 0 0 1 2 1 1 2 3 4 4 2 1 1 0
ϕ4,8 4 0 4 2 1 2 1 2 1 0 0 0 0 0 0 1 1 2 0 3 2 4 4 3 0 1 1
ϕ4,16 4 0 4 2 1 2 1 2 1 0 0 0 0 0 0 2 1 1 1 3 3 4 4 1 1 0 1
ϕ5,12 5 0 5 3 1 3 1 3 1 1 0 0 0 0 0 3 2 3 0 3 2 5 5 2 1 1 0
ϕ5,8 5 0 5 3 1 3 1 3 1 0 0 1 0 0 0 3 3 2 1 2 3 5 5 1 0 1 1
ϕ5,4 5 0 5 3 1 3 1 3 1 0 0 0 0 1 0 2 3 3 0 2 2 5 5 3 1 0 1
ϕ6,7 6 0 6 2 2 4 1 4 1 0 0 0 0 0 0 4 2 2 0 4 4 6 6 2 1 1 1
ϕ6,9 6 0 6 4 1 2 2 4 1 0 0 0 0 0 0 2 4 2 1 2 4 6 6 3 1 1 1
ϕ6,5 6 0 6 4 1 4 1 2 2 0 0 0 0 0 0 2 2 4 1 4 2 6 6 3 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 t40 t21 0 t11 0 t 0 t2 0 0 0 0 0 t3 0 t13 0 t8 0 t12 t8 0 0 0 0
ϕ1,20 t40 1 t20 t 0 t21 0 t11 0 0 0 t2 0 0 0 0 t3 0 0 0 t8 t4 t12 0 0 0 0
ϕ1,40 t20 0 1 t11 0 t 0 t21 0 0 0 0 0 t2 0 0 t13 t3 t8 0 0 t8 t4 0 0 0 0
ϕ2,21 t21 + t39 0 t + t19 1 + t12 0 t2 + t20 0 t10 + t22 0 0 0 t 0 t3 0 t12 t2 + t14 t4 0 0 t7 t3 + t9 t5 + t11 t7 0 0 0
ϕ2,27 t27 + t33 0 t7 + t13 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 t6 0 t10 0 t5 + t11 t t3 + t9 t5 + t11 t13 0 0 t2
ϕ2,11 t + t19 0 t11 + t29 t10 + t22 0 1 + t12 0 t2 + t20 0 t3 0 0 0 t 0 t4 t12 t2 + t14 t7 t9 0 t7 + t13 t3 + t9 0 0 0 0
ϕ2,17 t7 + t13 0 t17 + t23 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 t10 t6 0 t t15 t5 + t11 t7 + t13 t3 + t9 0 t2 0 0
ϕ2,1 t11 + t29 t t21 + t39 t2 + t20 0 t10 + t22 0 1 + t12 0 t 0 t3 0 0 0 t2 t4 t12 0 t7 t9 t5 + t11 t7 + t13 0 0 0 0
ϕ2,7 t17 + t23 0 t27 + t33 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 t10 t6 0 t t15 t5 + t11 t7 + t13 t3 + t9 0 t2 0
ϕ3,2 t10 + t22 + t28 0 t2 + t20 + t38 t + t13 + t19 0 t3 + t9 + t21 0 t11 t5 1 0 t2 0 t4 0 t + t13 t3 + t15 t5 + t11 0 t6 t8 t4 + 2t10 2t6 + t12 t8 0 0 0
ϕ3,14 t10 + t16 + t34 0 t14 + t20 + t26 t7 t7 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 t7 + t13 t3 + t9 0 0 t12 t2 + t8 + t14 t4 + 2t10 2t6 + t12 t2 0 0 0
ϕ3,10'' t2 + t20 + t38 0 t12 + t18 + t30 t11 t5 t + t13 + t19 0 t3 + t9 + t21 0 t4 0 1 0 t2 0 t5 + t11 t + t13 t3 + t15 0 t10 t6 t2 + t8 + t14 t4 + 2t10 t6 0 0 0
ϕ3,10' t14 + t20 + t26 0 t6 + t24 + t30 t5 + t11 + t17 0 t7 t7 t15 t3 0 0 0 1 0 0 0 t7 + t13 t3 + t9 t2 t4 t12 t2 + t8 + t14 t4 + 2t10 t6 + t12 0 0 0
ϕ3,12 t12 + t18 + t30 t2 t10 + t22 + t28 t3 + t9 + t21 0 t11 t5 t + t13 + t19 0 t2 0 t4 0 1 0 t3 t5 + t11 t + t13 t6 t8 t10 2t6 + t12 t2 + t8 + t14 0 0 0 0
ϕ3,6 t6 + t24 + t30 0 t10 + t16 + t34 t15 t3 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 t3 + t9 0 t7 + t13 0 t2 + t8 + t14 t4 2t6 + t12 t2 + t8 + t14 t10 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 0 t13 + t19 + t31 + t37 t12 + t18 t6 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 t 0 t3 0 1 + t6 + t12 t2 + t14 t4 + t10 + t16 0 t5 + t11 t7 t3 + 2t9 + t15 2t5 + 2t11 t7 0 0 t2
ϕ4,11 t13 + t19 + t31 + t37 t3 t11 + t17 + t23 + t29 t4 + t10 t4 t12 + t18 t6 t2 + t8 + t14 + t20 0 t3 0 0 0 t 0 t4 + t10 1 + t6 + t12 t2 + t14 0 t9 t5 + t11 t + 2t7 + t13 t3 + 2t9 + t15 t5 t2 0 0
ϕ4,13 t11 + t17 + t23 + t29 0 t3 + t9 + t21 + t27 t2 + t8 + t14 + t20 0 t4 + t10 t4 t12 + t18 t6 t 0 t3 0 0 0 t2 + t14 t4 + t10 + t16 1 + t6 + t12 t5 t7 t9 2t5 + 2t11 t + 2t7 + t13 t9 0 t2 0
ϕ4,6 t6 + t12 + t18 + t24 0 t16 + t22 + t28 + t34 t9 + t15 t9 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 t9 t5 + t11 t7 1 t2 + t14 t4 + t10 + t16 2t6 + 2t12 t2 + 2t8 + t14 t4 + t10 t t3 0
ϕ4,8 t16 + t22 + t28 + t34 0 t8 + t14 + t26 + t32 t7 + t13 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 t7 t9 t5 + t11 0 1 + t6 + t12 t2 + t14 2t4 + 2t10 2t6 + 2t12 t2 + t8 + t14 0 t t3
ϕ4,16 t8 + t14 + t26 + t32 0 t6 + t12 + t18 + t24 t5 + t17 t5 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 t5 + t11 t7 t9 t2 t4 + t10 + t16 1 + t6 + t12 t2 + 2t8 + t14 2t4 + 2t10 t12 t3 0 t
ϕ5,12 t12 + t18 + t24 + t30 + t36 0 t4 + t10 + t16 + t22 + t28 t3 + t9 + t15 t3 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t2 0 0 0 0 0 t3 + t9 + t15 t5 + t11 t + t7 + t13 0 t2 + t8 + t14 t4 + t10 1 + 2t6 + 2t12 t2 + 3t8 + t14 t4 + t10 t t3 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 0 t8 + t14 + t20 + t26 + t32 t7 + t13 + t19 t7 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 t2 0 0 0 t + t7 + t13 t3 + t9 + t15 t5 + t11 t4 t6 + t12 t2 + t8 + t14 2t4 + 2t10 + t16 1 + 2t6 + 2t12 t8 0 t t3
ϕ5,4 t8 + t14 + t20 + t26 + t32 0 t12 + t18 + t24 + t30 + t36 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 t5 + t11 t + t7 + t13 t3 + t9 + t15 0 t4 + t10 t6 + t12 t2 + 3t8 + t14 2t4 + 2t10 + t16 1 + t6 + t12 t3 0 t
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 0 t9 + 2t15 + t21 + t27 + t33 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 t6 + t12 0 t + t7 + 2t13 2t3 + t9 + t15 3t5 + 3t11 t + 3t7 + 2t13 t3 + t9 1 t2 t4
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 0 t7 + t13 + t19 + 2t25 + t31 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 t4 + t10 t3 t5 + t11 t + t7 + 2t13 2t3 + 3t9 + t15 3t5 + 3t11 t + t7 + t13 t4 1 t2
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 0 t5 + t11 + t17 + t23 + t29 + t35 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 t4 + t10 t6 + t12 t2 + 2t8 + t14 t 2t3 + t9 + t15 t5 + t11 t + 3t7 + 2t13 2t3 + 3t9 + t15 t5 + 2t11 t2 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,21,  ϕ2,27},   {ϕ4,3,  ϕ4,6},   {ϕ4,11,  ϕ1,20,  ϕ4,8,  ϕ2,11,  ϕ5,8,  ϕ2,17,  ϕ6,7,  ϕ6,9,  ϕ3,2,  ϕ6,5,  ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ3,6},   {ϕ2,1,  ϕ2,7},   {ϕ4,13,  ϕ4,16}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,20311 + 2t
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,11322 + t
ϕ2,17642 + 4t
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''1263 + 6t + 3t2
ϕ3,10'393
ϕ3,12363
ϕ3,6393
ϕ4,318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,119124 + 5t
ϕ4,1318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,8684 + 2t
ϕ4,1618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,89155 + 4t
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,721246 + 9t + 6t2
ϕ6,96246
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0
ϕ1,20 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0
ϕ1,40 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0
ϕ2,21 2 0 2 2 0 0 0 2 0 0 0 1 0 0 0 1 0 1 1 0 1 2 0 2 0 0 0
ϕ2,27 2 0 2 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 2 0 2 0 0 1
ϕ2,11 2 0 2 2 0 1 0 2 0 1 0 0 0 0 0 1 0 2 1 0 0 2 0 2 0 0 0
ϕ2,17 2 0 2 0 1 0 1 2 0 0 0 0 0 0 0 1 0 0 1 0 2 2 0 2 0 0 0
ϕ2,1 2 1 2 2 0 0 0 2 0 1 0 0 0 0 0 2 0 1 0 0 1 2 0 2 0 0 0
ϕ2,7 2 0 2 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 1 1 2 0 2 0 0 0
ϕ3,2 3 0 3 3 0 0 0 1 1 1 0 1 0 0 0 2 0 2 1 0 1 3 0 3 0 0 0
ϕ3,14 3 0 3 1 1 0 0 3 0 0 1 0 0 0 0 2 0 0 1 0 3 3 0 3 0 0 0
ϕ3,10'' 3 0 3 1 1 0 0 3 0 1 0 1 0 0 0 2 0 2 1 0 1 3 0 3 0 0 0
ϕ3,10' 3 0 3 3 0 0 0 1 1 0 0 0 1 0 0 0 0 2 3 0 1 3 0 3 0 0 0
ϕ3,12 3 0 3 3 0 0 0 3 0 1 0 0 0 1 0 2 0 2 1 0 1 3 0 3 0 0 0
ϕ3,6 3 0 3 1 1 0 0 1 1 0 0 0 0 0 1 2 0 2 1 0 1 3 0 3 0 0 0
ϕ4,3 4 0 4 2 1 0 0 2 1 0 0 1 0 0 0 3 0 3 1 0 1 4 0 4 0 0 1
ϕ4,11 4 0 4 2 1 0 0 4 0 1 0 0 0 0 0 3 1 2 1 0 2 4 0 4 0 0 0
ϕ4,13 4 0 4 4 0 0 0 2 1 1 0 0 0 0 0 2 0 3 2 0 1 4 1 4 0 0 0
ϕ4,6 4 0 4 2 1 0 0 2 1 0 0 0 0 0 0 1 0 1 3 0 3 4 0 4 1 0 0
ϕ4,8 4 0 4 2 1 0 0 2 1 0 0 0 0 0 0 1 0 2 3 1 2 4 0 4 0 0 1
ϕ4,16 4 0 4 2 1 0 1 2 1 0 0 0 0 0 0 2 0 1 2 0 3 4 0 4 0 0 1
ϕ5,12 5 0 5 3 1 0 0 3 1 1 0 0 0 0 0 3 0 3 2 0 2 5 0 5 1 0 0
ϕ5,8 5 0 5 3 1 0 0 3 1 0 0 0 0 0 0 3 0 2 2 0 3 5 1 5 0 0 1
ϕ5,4 5 0 5 3 1 0 0 3 1 0 0 0 0 0 0 2 1 3 3 0 2 5 0 5 0 0 1
ϕ6,7 6 0 6 2 2 0 0 4 1 0 0 0 0 0 0 4 0 2 2 0 4 6 0 6 1 0 1
ϕ6,9 6 0 6 4 1 0 0 4 1 0 0 0 0 0 0 2 0 2 4 0 4 6 0 6 0 1 1
ϕ6,5 6 0 6 4 1 0 0 2 2 0 0 0 0 0 0 2 0 4 4 0 2 6 0 6 1 0 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 t40 t21 0 0 0 t 0 t2 0 0 0 0 0 t3 0 t13 0 0 0 t12 0 t4 0 0 0
ϕ1,20 t40 1 t20 t 0 0 0 t11 0 0 0 0 0 0 0 t13 0 0 0 0 t8 t4 0 t8 0 0 0
ϕ1,40 t20 0 1 t11 0 t 0 t21 0 0 0 0 0 0 0 0 0 t3 t8 0 0 t8 0 t12 0 0 0
ϕ2,21 t21 + t39 0 t + t19 1 + t12 0 0 0 t10 + t22 0 0 0 t 0 0 0 t12 0 t4 t9 0 t7 t3 + t9 0 t7 + t13 0 0 0
ϕ2,27 t27 + t33 0 t7 + t13 0 1 0 0 0 t10 0 0 0 0 0 0 t6 0 t10 t15 0 t t3 + t9 0 t7 + t13 0 0 t2
ϕ2,11 t + t19 0 t11 + t29 t10 + t22 0 1 0 t2 + t20 0 t3 0 0 0 0 0 t4 0 t2 + t14 t7 0 0 t7 + t13 0 t5 + t11 0 0 0
ϕ2,17 t7 + t13 0 t17 + t23 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 t10 0 0 t 0 t5 + t11 t7 + t13 0 t5 + t11 0 0 0
ϕ2,1 t11 + t29 t t21 + t39 t2 + t20 0 0 0 1 + t12 0 t 0 0 0 0 0 t2 + t14 0 t12 0 0 t9 t5 + t11 0 t3 + t9 0 0 0
ϕ2,7 t17 + t23 0 t27 + t33 t8 + t14 0 0 0 0 1 0 0 0 0 0 0 0 0 t6 t5 + t11 t t15 t5 + t11 0 t3 + t9 0 0 0
ϕ3,2 t10 + t22 + t28 0 t2 + t20 + t38 t + t13 + t19 0 0 0 t11 t5 1 0 t2 0 0 0 t + t13 0 t5 + t11 t10 0 t8 t4 + 2t10 0 t2 + t8 + t14 0 0 0
ϕ3,14 t10 + t16 + t34 0 t14 + t20 + t26 t7 t7 0 0 t5 + t11 + t17 0 0 1 0 0 0 0 t7 + t13 0 0 t4 0 t2 + t8 + t14 t4 + 2t10 0 t2 + t8 + t14 0 0 0
ϕ3,10'' t2 + t20 + t38 0 t12 + t18 + t30 t11 t5 0 0 t3 + t9 + t21 0 t4 0 1 0 0 0 t5 + t11 0 t3 + t15 t8 0 t6 t2 + t8 + t14 0 2t6 + t12 0 0 0
ϕ3,10' t14 + t20 + t26 0 t6 + t24 + t30 t5 + t11 + t17 0 0 0 t15 t3 0 0 0 1 0 0 0 0 t3 + t9 t2 + t8 + t14 0 t12 t2 + t8 + t14 0 2t6 + t12 0 0 0
ϕ3,12 t12 + t18 + t30 0 t10 + t22 + t28 t3 + t9 + t21 0 0 0 t + t13 + t19 0 t2 0 0 0 1 0 t3 + t15 0 t + t13 t6 0 t10 2t6 + t12 0 t4 + 2t10 0 0 0
ϕ3,6 t6 + t24 + t30 0 t10 + t16 + t34 t15 t3 0 0 t7 t7 0 0 0 0 0 1 t3 + t9 0 t7 + t13 t12 0 t4 2t6 + t12 0 t4 + 2t10 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 0 t13 + t19 + t31 + t37 t12 + t18 t6 0 0 t4 + t10 t4 0 0 t 0 0 0 1 + t6 + t12 0 t4 + t10 + t16 t9 0 t7 t3 + 2t9 + t15 0 t + 2t7 + t13 0 0 t2
ϕ4,11 t13 + t19 + t31 + t37 0 t11 + t17 + t23 + t29 t4 + t10 t4 0 0 t2 + t8 + t14 + t20 0 t3 0 0 0 0 0 t4 + t10 + t16 1 t2 + t14 t7 0 t5 + t11 t + 2t7 + t13 0 2t5 + 2t11 0 0 0
ϕ4,13 t11 + t17 + t23 + t29 0 t3 + t9 + t21 + t27 t2 + t8 + t14 + t20 0 0 0 t12 + t18 t6 t 0 0 0 0 0 t2 + t14 0 1 + t6 + t12 t5 + t11 0 t9 2t5 + 2t11 t t3 + 2t9 + t15 0 0 0
ϕ4,6 t6 + t12 + t18 + t24 0 t16 + t22 + t28 + t34 t9 + t15 t9 0 0 t7 + t13 t 0 0 0 0 0 0 t9 0 t7 1 + t6 + t12 0 t4 + t10 + t16 2t6 + 2t12 0 2t4 + 2t10 t 0 0
ϕ4,8 t16 + t22 + t28 + t34 0 t8 + t14 + t26 + t32 t7 + t13 t 0 0 t5 + t17 t5 0 0 0 0 0 0 t7 0 t5 + t11 t4 + t10 + t16 1 t2 + t14 2t4 + 2t10 0 t2 + 2t8 + t14 0 0 t3
ϕ4,16 t8 + t14 + t26 + t32 0 t6 + t12 + t18 + t24 t5 + t17 t5 0 t t9 + t15 t9 0 0 0 0 0 0 t5 + t11 0 t9 t2 + t14 0 1 + t6 + t12 t2 + 2t8 + t14 0 2t6 + 2t12 0 0 t
ϕ5,12 t12 + t18 + t24 + t30 + t36 0 t4 + t10 + t16 + t22 + t28 t3 + t9 + t15 t3 0 0 t7 + t13 + t19 t7 t2 0 0 0 0 0 t3 + t9 + t15 0 t + t7 + t13 t6 + t12 0 t4 + t10 1 + 2t6 + 2t12 0 2t4 + 2t10 + t16 t 0 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 0 t8 + t14 + t20 + t26 + t32 t7 + t13 + t19 t7 0 0 t5 + t11 + t17 t5 0 0 0 0 0 0 t + t7 + t13 0 t5 + t11 t4 + t10 0 t2 + t8 + t14 2t4 + 2t10 + t16 1 t2 + 3t8 + t14 0 0 t3
ϕ5,4 t8 + t14 + t20 + t26 + t32 0 t12 + t18 + t24 + t30 + t36 t5 + t11 + t17 t5 0 0 t3 + t9 + t15 t3 0 0 0 0 0 0 t5 + t11 t t3 + t9 + t15 t2 + t8 + t14 0 t6 + t12 t2 + 3t8 + t14 0 1 + 2t6 + 2t12 0 0 t
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 0 t9 + 2t15 + t21 + t27 + t33 t8 + t14 t2 + t8 0 0 2t6 + t12 + t18 t6 0 0 0 0 0 0 t2 + 2t8 + t14 0 t6 + t12 t5 + t11 0 2t3 + t9 + t15 3t5 + 3t11 0 2t3 + 3t9 + t15 1 0 t4
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 0 t7 + t13 + t19 + 2t25 + t31 2t6 + t12 + t18 t6 0 0 t4 + t10 + 2t16 t4 0 0 0 0 0 0 t6 + t12 0 t4 + t10 2t3 + t9 + t15 0 t + t7 + 2t13 2t3 + 3t9 + t15 0 t + 3t7 + 2t13 0 1 t2
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 0 t5 + t11 + t17 + t23 + t29 + t35 t4 + t10 + 2t16 t4 0 0 t8 + t14 t2 + t8 0 0 0 0 0 0 t4 + t10 0 t2 + 2t8 + t14 t + t7 + 2t13 0 t5 + t11 t + 3t7 + 2t13 0 3t5 + 3t11 t2 0 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,3,  ϕ2,11,  ϕ2,17,  ϕ2,1,  ϕ4,6,  ϕ2,7},   {ϕ4,11,  ϕ4,13,  ϕ4,8,  ϕ4,16},   {ϕ1,20,  ϕ1,40,  ϕ2,21,  ϕ2,27},   {ϕ5,8,  ϕ5,4},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,2012111 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 10t11 + 9t12 + 8t13 + 7t14 + 6t15 + 5t16 + 4t17 + 3t18 + 2t19 + t20
ϕ1,40111
ϕ2,214742 + 3t + 4t2 + 5t3 + 6t4 + 7t5 + 6t6 + 5t7 + 4t8 + 3t9 + 2t10
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,11822 + 4t + 2t2
ϕ2,174242 + 4t + 6t2 + 8t3 + 10t4 + 8t5 + 4t6
ϕ2,19622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 10t8 + 8t9 + 6t10 + 4t11 + 2t12
ϕ2,7242
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,34284 + 8t + 10t2 + 8t3 + 6t4 + 4t5 + 2t6
ϕ4,1112044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 8t10 + 4t11
ϕ4,136044 + 8t + 12t2 + 12t3 + 12t4 + 8t5 + 4t6
ϕ4,628124 + 6t + 8t2 + 6t3 + 4t4
ϕ4,812044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 8t10 + 4t11
ϕ4,166044 + 8t + 12t2 + 12t3 + 12t4 + 8t5 + 4t6
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,812055 + 10t + 15t2 + 20t3 + 20t4 + 20t5 + 15t6 + 10t7 + 5t8
ϕ5,424055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 25t8 + 20t9 + 15t10 + 10t11 + 5t12
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0
ϕ1,20 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0
ϕ1,40 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0
ϕ2,21 2 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 0 0 2 1 1 0 0 0
ϕ2,27 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 2 1 1 0 0 1
ϕ2,11 2 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 2 1 1 0 0 0
ϕ2,17 2 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 1 1 1 0 0
ϕ2,1 2 2 0 0 0 0 0 2 0 1 0 1 0 0 0 0 1 0 0 1 0 2 0 2 0 0 0
ϕ2,7 2 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 2 0 2 0 1 0
ϕ3,2 3 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 3 1 2 0 0 0
ϕ3,14 3 1 0 0 1 0 1 1 0 0 1 0 0 0 0 0 2 0 0 0 1 3 1 2 0 0 0
ϕ3,10'' 3 1 0 0 1 1 0 2 0 1 0 1 0 1 0 0 1 1 0 1 0 3 1 2 0 0 0
ϕ3,10' 3 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 3 1 2 0 0 0
ϕ3,12 3 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 1 0 3 1 2 0 0 0
ϕ3,6 3 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 2 1 3 1 2 0 0 0
ϕ4,3 4 2 0 0 1 0 0 2 0 0 0 1 0 1 0 1 1 1 0 2 0 4 1 3 0 0 1
ϕ4,11 4 2 0 0 1 0 1 2 0 1 0 0 0 1 0 0 2 1 0 1 0 4 1 3 1 0 0
ϕ4,13 4 0 0 2 0 0 1 0 0 1 0 1 0 0 0 1 1 2 0 1 0 4 2 2 0 1 0
ϕ4,6 4 2 0 0 1 0 0 1 0 0 0 0 0 0 0 0 2 0 1 1 1 4 1 3 1 1 0
ϕ4,8 4 2 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 2 1 4 1 3 0 1 1
ϕ4,16 4 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 2 4 2 2 1 0 1
ϕ5,12 5 1 0 1 1 0 1 1 0 1 0 0 0 0 0 1 1 1 0 2 1 5 2 3 1 1 0
ϕ5,8 5 1 0 1 1 0 1 1 0 0 0 1 0 0 0 1 2 1 0 1 1 5 2 3 0 1 1
ϕ5,4 5 3 0 0 1 0 0 2 0 0 0 0 0 1 0 0 2 1 1 2 0 5 1 4 1 0 1
ϕ6,7 6 2 0 0 2 0 1 2 0 0 0 0 0 0 0 1 2 0 0 2 2 6 2 4 1 1 1
ϕ6,9 6 2 0 1 1 0 1 1 0 0 0 0 0 0 0 0 3 1 1 1 1 6 2 4 1 1 1
ϕ6,5 6 2 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 1 3 1 6 2 4 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 0 0 0 0 0 t 0 t2 0 0 0 0 0 0 0 0 0 t8 0 t12 0 t4 0 0 0
ϕ1,20 t40 1 0 0 0 0 0 t11 0 0 0 t2 0 0 0 0 t3 0 0 0 0 t4 0 t8 0 0 0
ϕ1,40 t20 0 1 0 0 t 0 0 0 0 0 0 0 t2 0 0 0 t3 0 0 0 t8 t4 0 0 0 0
ϕ2,21 t21 + t39 0 0 1 0 t2 0 t10 0 0 0 t 0 t3 0 0 t2 t4 0 0 0 t3 + t9 t5 t7 0 0 0
ϕ2,27 t27 + t33 0 0 0 1 0 0 0 0 0 0 0 0 0 0 t6 0 0 0 t5 t t3 + t9 t5 t7 0 0 t2
ϕ2,11 t + t19 0 0 t10 0 1 0 t2 0 t3 0 0 0 t 0 0 0 t2 0 t9 0 t7 + t13 t3 t5 0 0 0
ϕ2,17 t7 + t13 0 0 0 t10 0 1 0 0 0 0 0 0 0 0 0 t6 0 0 0 t5 t7 + t13 t3 t5 t2 0 0
ϕ2,1 t11 + t29 t + t19 0 0 0 0 0 1 + t12 0 t 0 t3 0 0 0 0 t4 0 0 t7 0 t5 + t11 0 t3 + t9 0 0 0
ϕ2,7 t17 + t23 t7 + t13 0 0 0 0 0 0 1 0 0 0 0 0 0 0 t10 0 0 t 0 t5 + t11 0 t3 + t9 0 t2 0
ϕ3,2 t10 + t22 + t28 t18 0 t 0 0 0 t11 0 1 0 t2 0 t4 0 t t3 t5 0 t6 0 t4 + 2t10 t6 t2 + t8 0 0 0
ϕ3,14 t10 + t16 + t34 t6 0 0 t7 0 t3 t5 0 0 1 0 0 0 0 0 t3 + t9 0 0 0 t2 t4 + 2t10 t6 t2 + t8 0 0 0
ϕ3,10'' t2 + t20 + t38 t10 0 0 t5 t 0 t3 + t9 0 t4 0 1 0 t2 0 0 t t3 0 t10 0 t2 + t8 + t14 t4 2t6 0 0 0
ϕ3,10' t14 + t20 + t26 t10 0 t5 0 0 0 0 0 0 0 0 1 0 0 0 t7 t3 t2 t4 0 t2 + t8 + t14 t4 2t6 0 0 0
ϕ3,12 t12 + t18 + t30 t2 0 t9 0 0 t5 t 0 t2 0 t4 0 1 0 0 t5 t 0 t8 0 2t6 + t12 t2 t4 + t10 0 0 0
ϕ3,6 t6 + t24 + t30 t14 0 0 t3 0 0 t7 0 0 0 0 0 0 1 t3 0 0 0 t2 + t8 t4 2t6 + t12 t2 t4 + t10 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 t11 + t17 0 0 t6 0 0 t4 + t10 0 0 0 t 0 t3 0 1 t2 t4 0 t5 + t11 0 t3 + 2t9 + t15 t5 t + 2t7 0 0 t2
ϕ4,11 t13 + t19 + t31 + t37 t3 + t9 0 0 t4 0 t6 t2 + t8 0 t3 0 0 0 t 0 0 1 + t6 t2 0 t9 0 t + 2t7 + t13 t3 2t5 + t11 t2 0 0
ϕ4,13 t11 + t17 + t23 + t29 0 0 t2 + t8 0 0 t4 0 0 t 0 t3 0 0 0 t2 t4 1 + t6 0 t7 0 2t5 + 2t11 t + t7 t3 + t9 0 t2 0
ϕ4,6 t6 + t12 + t18 + t24 t8 + t14 0 0 t9 0 0 t7 0 0 0 0 0 0 0 0 t5 + t11 0 1 t2 t4 2t6 + 2t12 t2 2t4 + t10 t t3 0
ϕ4,8 t16 + t22 + t28 + t34 t6 + t12 0 0 t 0 0 t5 0 0 0 0 0 0 0 0 t9 0 t4 1 + t6 t2 2t4 + 2t10 t6 t2 + 2t8 0 t t3
ϕ4,16 t8 + t14 + t26 + t32 0 0 t5 t5 0 t 0 0 0 0 0 0 0 0 t5 t7 0 0 t4 1 + t6 t2 + 2t8 + t14 2t4 2t6 t3 0 t
ϕ5,12 t12 + t18 + t24 + t30 + t36 t8 0 t3 t3 0 t5 t7 0 t2 0 0 0 0 0 t3 t5 t 0 t2 + t8 t4 1 + 2t6 + 2t12 t2 + t8 2t4 + t10 t t3 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 t12 0 t7 t7 0 t3 t5 0 0 0 t2 0 0 0 t t3 + t9 t5 0 t6 t2 2t4 + 2t10 + t16 1 + t6 t2 + 2t8 0 t t3
ϕ5,4 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 0 0 t5 0 0 t3 + t9 0 0 0 0 0 t2 0 0 t + t7 t3 t2 t4 + t10 0 t2 + 3t8 + t14 t4 1 + 2t6 + t12 t3 0 t
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 0 0 t2 + t8 0 t4 2t6 0 0 0 0 0 0 0 t2 t4 + t10 0 0 t + t7 2t3 3t5 + 3t11 t + t7 2t3 + 2t9 1 t2 t4
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 t5 + t11 0 t6 t6 0 t2 t4 0 0 0 0 0 0 0 0 t2 + 2t8 t4 t3 t5 t 2t3 + 3t9 + t15 2t5 t + 3t7 t4 1 t2
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 t9 + t15 0 t4 t4 0 0 t8 0 0 0 0 0 0 0 t4 t6 t2 t 2t3 + t9 t5 t + 3t7 + 2t13 2t3 3t5 + t11 t2 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ2,7},   {ϕ1,0,  ϕ4,6,  ϕ2,21,  ϕ2,27,  ϕ5,12,  ϕ6,7,  ϕ6,9,  ϕ3,2,  ϕ6,5,  ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ3,6,  ϕ4,3},   {ϕ4,11,  ϕ4,8},   {ϕ2,11,  ϕ2,17},   {ϕ4,13,  ϕ4,16}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0311 + 2t
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,21322 + t
ϕ2,27642 + 4t
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,21263 + 6t + 3t2
ϕ3,14393
ϕ3,10''363
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,39124 + 5t
ϕ4,1118044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,6684 + 2t
ϕ4,818044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ5,129155 + 4t
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,76246
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,521246 + 9t + 6t2

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0
ϕ1,20 0 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0
ϕ1,40 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0
ϕ2,21 0 2 2 1 0 2 0 2 0 0 0 0 0 1 0 0 2 1 0 0 1 0 2 2 0 0 0
ϕ2,27 0 2 2 0 1 2 0 0 1 0 0 0 0 0 0 0 0 1 0 2 1 0 2 2 0 0 0
ϕ2,11 1 2 2 0 0 2 0 2 0 0 0 0 0 1 0 0 1 2 0 1 0 0 2 2 0 0 0
ϕ2,17 0 2 2 0 0 0 1 2 0 0 0 0 0 0 0 0 1 0 1 1 2 0 2 2 0 0 0
ϕ2,1 0 2 2 0 0 2 0 2 0 1 0 0 0 0 0 0 1 1 0 1 1 0 2 2 0 0 0
ϕ2,7 0 2 2 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 2 2 0 1 0
ϕ3,2 0 3 3 0 0 3 0 1 1 1 0 0 0 1 0 0 2 2 0 1 1 0 3 3 0 0 0
ϕ3,14 0 3 3 0 0 1 1 3 0 0 1 0 0 0 0 0 2 0 0 1 3 0 3 3 0 0 0
ϕ3,10'' 0 3 3 0 0 3 0 3 0 0 0 1 0 1 0 0 2 2 0 1 1 0 3 3 0 0 0
ϕ3,10' 0 3 3 0 0 1 1 1 1 0 0 0 1 0 0 0 2 2 0 1 1 0 3 3 0 0 0
ϕ3,12 0 3 3 0 0 1 1 3 0 1 0 0 0 1 0 0 2 2 0 1 1 0 3 3 0 0 0
ϕ3,6 0 3 3 0 0 3 0 1 1 0 0 0 0 0 1 0 0 2 0 3 1 0 3 3 0 0 0
ϕ4,3 0 4 4 0 0 4 0 2 1 0 0 0 0 1 0 1 2 3 0 2 1 0 4 4 0 0 0
ϕ4,11 0 4 4 0 0 2 1 4 0 0 0 0 0 1 0 0 3 2 0 1 2 1 4 4 0 0 0
ϕ4,13 0 4 4 0 0 2 1 2 1 1 0 0 0 0 0 0 3 3 0 1 1 0 4 4 0 1 0
ϕ4,6 0 4 4 0 0 2 1 2 1 0 0 0 0 0 0 0 2 1 1 2 3 0 4 4 0 1 0
ϕ4,8 0 4 4 0 1 2 1 2 1 0 0 0 0 0 0 0 1 2 0 3 2 0 4 4 0 1 0
ϕ4,16 0 4 4 0 0 2 1 2 1 0 0 0 0 0 0 0 1 1 0 3 3 0 4 4 0 0 1
ϕ5,12 0 5 5 0 0 3 1 3 1 0 0 0 0 0 0 0 2 3 0 3 2 1 5 5 0 1 0
ϕ5,8 0 5 5 0 0 3 1 3 1 0 0 0 0 0 0 1 3 2 0 2 3 0 5 5 0 1 0
ϕ5,4 0 5 5 0 0 3 1 3 1 0 0 0 0 1 0 0 3 3 0 2 2 0 5 5 0 0 1
ϕ6,7 0 6 6 0 0 4 1 4 1 0 0 0 0 0 0 0 2 2 0 4 4 0 6 6 1 1 0
ϕ6,9 0 6 6 0 0 2 2 4 1 0 0 0 0 0 0 0 4 2 0 2 4 0 6 6 0 1 1
ϕ6,5 0 6 6 0 0 4 1 2 2 0 0 0 0 0 0 0 2 4 0 4 2 0 6 6 0 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 t40 0 0 t11 0 t 0 0 0 0 0 0 0 0 0 t13 0 t8 0 0 t8 t4 0 0 0
ϕ1,20 0 1 t20 t 0 t21 0 t11 0 0 0 0 0 0 0 0 t3 0 0 0 t8 0 t12 t8 0 0 0
ϕ1,40 0 t40 1 0 0 t 0 t21 0 0 0 0 0 t2 0 0 t13 t3 0 0 0 0 t4 t12 0 0 0
ϕ2,21 0 t11 + t29 t + t19 1 0 t2 + t20 0 t10 + t22 0 0 0 0 0 t3 0 0 t2 + t14 t4 0 0 t7 0 t5 + t11 t7 + t13 0 0 0
ϕ2,27 0 t17 + t23 t7 + t13 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 0 0 t10 0 t5 + t11 t 0 t5 + t11 t7 + t13 0 0 0
ϕ2,11 t t21 + t39 t11 + t29 0 0 1 + t12 0 t2 + t20 0 0 0 0 0 t 0 0 t12 t2 + t14 0 t9 0 0 t3 + t9 t5 + t11 0 0 0
ϕ2,17 0 t27 + t33 t17 + t23 0 0 0 1 t8 + t14 0 0 0 0 0 0 0 0 t6 0 t t15 t5 + t11 0 t3 + t9 t5 + t11 0 0 0
ϕ2,1 0 t + t19 t21 + t39 0 0 t10 + t22 0 1 + t12 0 t 0 0 0 0 0 0 t4 t12 0 t7 t9 0 t7 + t13 t3 + t9 0 0 0
ϕ2,7 0 t7 + t13 t27 + t33 0 0 0 t10 0 1 0 0 0 0 0 0 0 t10 t6 0 t t15 0 t7 + t13 t3 + t9 0 t2 0
ϕ3,2 0 t12 + t18 + t30 t2 + t20 + t38 0 0 t3 + t9 + t21 0 t11 t5 1 0 0 0 t4 0 0 t3 + t15 t5 + t11 0 t6 t8 0 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,14 0 t6 + t24 + t30 t14 + t20 + t26 0 0 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 0 t3 + t9 0 0 t12 t2 + t8 + t14 0 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,10'' 0 t10 + t22 + t28 t12 + t18 + t30 0 0 t + t13 + t19 0 t3 + t9 + t21 0 0 0 1 0 t2 0 0 t + t13 t3 + t15 0 t10 t6 0 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,10' 0 t10 + t16 + t34 t6 + t24 + t30 0 0 t7 t7 t15 t3 0 0 0 1 0 0 0 t7 + t13 t3 + t9 0 t4 t12 0 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,12 0 t2 + t20 + t38 t10 + t22 + t28 0 0 t11 t5 t + t13 + t19 0 t2 0 0 0 1 0 0 t5 + t11 t + t13 0 t8 t10 0 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ3,6 0 t14 + t20 + t26 t10 + t16 + t34 0 0 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 0 0 t7 + t13 0 t2 + t8 + t14 t4 0 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ4,3 0 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 0 0 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 0 0 t3 0 1 t2 + t14 t4 + t10 + t16 0 t5 + t11 t7 0 2t5 + 2t11 t + 2t7 + t13 0 0 0
ϕ4,11 0 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 0 0 t12 + t18 t6 t2 + t8 + t14 + t20 0 0 0 0 0 t 0 0 1 + t6 + t12 t2 + t14 0 t9 t5 + t11 t t3 + 2t9 + t15 2t5 + 2t11 0 0 0
ϕ4,13 0 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 0 0 t4 + t10 t4 t12 + t18 t6 t 0 0 0 0 0 0 t4 + t10 + t16 1 + t6 + t12 0 t7 t9 0 t + 2t7 + t13 t3 + 2t9 + t15 0 t2 0
ϕ4,6 0 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 0 0 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 0 t5 + t11 t7 1 t2 + t14 t4 + t10 + t16 0 t2 + 2t8 + t14 2t4 + 2t10 0 t3 0
ϕ4,8 0 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 0 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 0 t9 t5 + t11 0 1 + t6 + t12 t2 + t14 0 2t6 + 2t12 t2 + 2t8 + t14 0 t 0
ϕ4,16 0 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 0 0 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 0 t7 t9 0 t4 + t10 + t16 1 + t6 + t12 0 2t4 + 2t10 2t6 + 2t12 0 0 t
ϕ5,12 0 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 0 0 t5 + t11 + t17 t5 t7 + t13 + t19 t7 0 0 0 0 0 0 0 t5 + t11 t + t7 + t13 0 t2 + t8 + t14 t4 + t10 1 t2 + 3t8 + t14 2t4 + 2t10 + t16 0 t3 0
ϕ5,8 0 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 0 0 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 0 0 0 0 t t3 + t9 + t15 t5 + t11 0 t6 + t12 t2 + t8 + t14 0 1 + 2t6 + 2t12 t2 + 3t8 + t14 0 t 0
ϕ5,4 0 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 0 0 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 0 t + t7 + t13 t3 + t9 + t15 0 t4 + t10 t6 + t12 0 2t4 + 2t10 + t16 1 + 2t6 + 2t12 0 0 t
ϕ6,7 0 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 0 0 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 0 t4 + t10 t6 + t12 0 t + t7 + 2t13 2t3 + t9 + t15 0 t + 3t7 + 2t13 2t3 + 3t9 + t15 1 t2 0
ϕ6,9 0 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 0 0 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 0 t5 + t11 t + t7 + 2t13 0 3t5 + 3t11 t + 3t7 + 2t13 0 1 t2
ϕ6,5 0 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 0 0 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 0 2t3 + t9 + t15 t5 + t11 0 2t3 + 3t9 + t15 3t5 + 3t11 0 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ5,4,  ϕ4,11,  ϕ4,8},   {ϕ2,11,  ϕ2,17},   {ϕ2,21,  ϕ2,27},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2},   {ϕ4,3,  ϕ4,6},   {ϕ2,1,  ϕ2,7},   {ϕ4,13,  ϕ4,16}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,04011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 4t8
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,111044 + 3t + 2t2 + t3
ϕ4,1318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,814044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 11t8 + 10t9 + 9t10 + 8t11 + 7t12 + 6t13 + 5t14
ϕ4,1618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,417055 + 10t + 11t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 8t14 + 4t15
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0
ϕ1,20 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0
ϕ1,40 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0
ϕ2,21 0 2 2 2 0 2 0 2 0 0 0 1 0 1 0 1 1 1 1 0 1 2 2 1 0 0 0
ϕ2,27 0 2 2 0 1 2 0 0 1 0 0 0 0 0 0 1 0 1 1 2 1 2 2 0 0 0 1
ϕ2,11 1 2 2 2 0 2 0 2 0 1 0 0 0 1 0 1 0 2 1 0 0 2 2 1 0 0 0
ϕ2,17 1 2 2 0 1 0 1 2 0 0 0 0 0 0 0 1 0 0 1 0 2 2 2 1 1 0 0
ϕ2,1 0 2 2 2 0 2 0 2 0 1 0 1 0 0 0 2 0 1 0 1 1 2 2 1 0 0 0
ϕ2,7 0 2 2 2 0 0 1 0 1 0 0 0 0 0 0 0 0 1 2 1 1 2 2 1 0 1 0
ϕ3,2 0 3 3 3 0 3 0 1 1 1 0 1 0 1 0 2 0 2 1 1 1 3 3 2 0 0 0
ϕ3,14 0 3 3 1 1 1 1 3 0 0 1 0 0 0 0 2 0 0 1 1 3 3 3 2 0 0 0
ϕ3,10'' 1 3 3 1 1 3 0 3 0 1 0 1 0 1 0 2 1 2 1 0 1 3 3 1 0 0 0
ϕ3,10' 0 3 3 3 0 1 1 1 1 0 0 0 1 0 0 0 0 2 3 1 1 3 3 2 0 0 0
ϕ3,12 0 3 3 3 0 1 1 3 0 1 0 1 0 1 0 2 0 2 1 1 1 3 3 2 0 0 0
ϕ3,6 1 3 3 1 1 3 0 1 1 0 0 0 0 0 1 2 0 2 1 2 1 3 3 0 0 0 0
ϕ4,3 1 4 4 2 1 4 0 2 1 0 0 1 0 1 0 3 0 3 1 1 1 4 4 2 0 0 1
ϕ4,11 0 4 4 2 1 2 1 4 0 1 0 0 0 1 0 3 1 2 1 1 2 4 4 2 1 0 0
ϕ4,13 0 4 4 4 0 2 1 2 1 1 0 1 0 0 0 2 0 3 2 1 1 4 4 3 0 1 0
ϕ4,6 1 4 4 2 1 2 1 2 1 0 0 0 0 0 0 1 0 1 3 1 3 4 4 2 1 1 0
ϕ4,8 0 4 4 2 1 2 1 2 1 0 0 0 0 0 0 1 0 2 3 3 2 4 4 1 0 1 1
ϕ4,16 1 4 4 2 1 2 1 2 1 0 0 0 0 0 0 2 0 1 2 2 3 4 4 1 1 0 1
ϕ5,12 0 5 5 3 1 3 1 3 1 1 0 0 0 0 0 3 0 3 2 3 2 5 5 2 1 1 0
ϕ5,8 1 5 5 3 1 3 1 3 1 0 0 1 0 0 0 3 0 2 2 1 3 5 5 3 0 1 1
ϕ5,4 0 5 5 3 1 3 1 3 1 0 0 0 0 1 0 2 0 3 3 2 2 5 5 3 1 0 1
ϕ6,7 1 6 6 2 2 4 1 4 1 0 0 0 0 0 0 4 0 2 2 3 4 6 6 2 1 1 1
ϕ6,9 0 6 6 4 1 2 2 4 1 0 0 0 0 0 0 2 0 2 4 2 4 6 6 4 1 1 1
ϕ6,5 1 6 6 4 1 4 1 2 2 0 0 0 0 0 0 2 0 4 4 3 2 6 6 2 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 t40 t21 0 t11 0 t 0 t2 0 0 0 0 0 t3 0 t13 0 0 0 t12 t8 0 0 0 0
ϕ1,20 0 1 t20 t 0 t21 0 t11 0 0 0 t2 0 0 0 t13 t3 0 0 0 t8 t4 t12 0 0 0 0
ϕ1,40 0 t40 1 t11 0 t 0 t21 0 0 0 0 0 t2 0 0 0 t3 t8 0 0 t8 t4 t12 0 0 0
ϕ2,21 0 t11 + t29 t + t19 1 + t12 0 t2 + t20 0 t10 + t22 0 0 0 t 0 t3 0 t12 t2 t4 t9 0 t7 t3 + t9 t5 + t11 t13 0 0 0
ϕ2,27 0 t17 + t23 t7 + t13 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 t6 0 t10 t15 t5 + t11 t t3 + t9 t5 + t11 0 0 0 t2
ϕ2,11 t t21 + t39 t11 + t29 t10 + t22 0 1 + t12 0 t2 + t20 0 t3 0 0 0 t 0 t4 0 t2 + t14 t7 0 0 t7 + t13 t3 + t9 t11 0 0 0
ϕ2,17 t7 t27 + t33 t17 + t23 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 t10 0 0 t 0 t5 + t11 t7 + t13 t3 + t9 t5 t2 0 0
ϕ2,1 0 t + t19 t21 + t39 t2 + t20 0 t10 + t22 0 1 + t12 0 t 0 t3 0 0 0 t2 + t14 0 t12 0 t7 t9 t5 + t11 t7 + t13 t3 0 0 0
ϕ2,7 0 t7 + t13 t27 + t33 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 0 t6 t5 + t11 t t15 t5 + t11 t7 + t13 t9 0 t2 0
ϕ3,2 0 t12 + t18 + t30 t2 + t20 + t38 t + t13 + t19 0 t3 + t9 + t21 0 t11 t5 1 0 t2 0 t4 0 t + t13 0 t5 + t11 t10 t6 t8 t4 + 2t10 2t6 + t12 t2 + t14 0 0 0
ϕ3,14 0 t6 + t24 + t30 t14 + t20 + t26 t7 t7 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 t7 + t13 0 0 t4 t12 t2 + t8 + t14 t4 + 2t10 2t6 + t12 t2 + t8 0 0 0
ϕ3,10'' t2 t10 + t22 + t28 t12 + t18 + t30 t11 t5 t + t13 + t19 0 t3 + t9 + t21 0 t4 0 1 0 t2 0 t5 + t11 t t3 + t15 t8 0 t6 t2 + t8 + t14 t4 + 2t10 t12 0 0 0
ϕ3,10' 0 t10 + t16 + t34 t6 + t24 + t30 t5 + t11 + t17 0 t7 t7 t15 t3 0 0 0 1 0 0 0 0 t3 + t9 t2 + t8 + t14 t4 t12 t2 + t8 + t14 t4 + 2t10 t6 + t12 0 0 0
ϕ3,12 0 t2 + t20 + t38 t10 + t22 + t28 t3 + t9 + t21 0 t11 t5 t + t13 + t19 0 t2 0 t4 0 1 0 t3 + t15 0 t + t13 t6 t8 t10 2t6 + t12 t2 + t8 + t14 t4 + t10 0 0 0
ϕ3,6 t6 t14 + t20 + t26 t10 + t16 + t34 t15 t3 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 t3 + t9 0 t7 + t13 t12 t2 + t8 t4 2t6 + t12 t2 + t8 + t14 0 0 0 0
ϕ4,3 t3 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 t12 + t18 t6 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 t 0 t3 0 1 + t6 + t12 0 t4 + t10 + t16 t9 t5 t7 t3 + 2t9 + t15 2t5 + 2t11 t + t13 0 0 t2
ϕ4,11 0 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 t4 + t10 t4 t12 + t18 t6 t2 + t8 + t14 + t20 0 t3 0 0 0 t 0 t4 + t10 + t16 1 t2 + t14 t7 t9 t5 + t11 t + 2t7 + t13 t3 + 2t9 + t15 t5 + t11 t2 0 0
ϕ4,13 0 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 t2 + t8 + t14 + t20 0 t4 + t10 t4 t12 + t18 t6 t 0 t3 0 0 0 t2 + t14 0 1 + t6 + t12 t5 + t11 t7 t9 2t5 + 2t11 t + 2t7 + t13 t3 + t9 + t15 0 t2 0
ϕ4,6 t6 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 t9 + t15 t9 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 t9 0 t7 1 + t6 + t12 t2 t4 + t10 + t16 2t6 + 2t12 t2 + 2t8 + t14 t4 + t10 t t3 0
ϕ4,8 0 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 t7 + t13 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 t7 0 t5 + t11 t4 + t10 + t16 1 + t6 + t12 t2 + t14 2t4 + 2t10 2t6 + 2t12 t8 0 t t3
ϕ4,16 t8 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 t5 + t17 t5 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 t5 + t11 0 t9 t2 + t14 t4 + t10 1 + t6 + t12 t2 + 2t8 + t14 2t4 + 2t10 t6 t3 0 t
ϕ5,12 0 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 t3 + t9 + t15 t3 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t2 0 0 0 0 0 t3 + t9 + t15 0 t + t7 + t13 t6 + t12 t2 + t8 + t14 t4 + t10 1 + 2t6 + 2t12 t2 + 3t8 + t14 t4 + t10 t t3 0
ϕ5,8 t4 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 t7 + t13 + t19 t7 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 t2 0 0 0 t + t7 + t13 0 t5 + t11 t4 + t10 t6 t2 + t8 + t14 2t4 + 2t10 + t16 1 + 2t6 + 2t12 t2 + t8 + t14 0 t t3
ϕ5,4 0 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 t5 + t11 0 t3 + t9 + t15 t2 + t8 + t14 t4 + t10 t6 + t12 t2 + 3t8 + t14 2t4 + 2t10 + t16 1 + t6 + t12 t3 0 t
ϕ6,7 t5 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 t2 + 2t8 + t14 0 t6 + t12 t5 + t11 t + t7 + t13 2t3 + t9 + t15 3t5 + 3t11 t + 3t7 + 2t13 t3 + t9 1 t2 t4
ϕ6,9 0 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 t6 + t12 0 t4 + t10 2t3 + t9 + t15 t5 + t11 t + t7 + 2t13 2t3 + 3t9 + t15 3t5 + 3t11 t + 2t7 + t13 t4 1 t2
ϕ6,5 t7 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 t4 + t10 0 t2 + 2t8 + t14 t + t7 + 2t13 2t3 + t9 t5 + t11 t + 3t7 + 2t13 2t3 + 3t9 + t15 t5 + t11 t2 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ5,12,  ϕ1,20,  ϕ4,13,  ϕ4,16},   {ϕ2,11,  ϕ2,17},   {ϕ2,21,  ϕ2,27},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2},   {ϕ4,11,  ϕ4,8},   {ϕ4,3,  ϕ4,6},   {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)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,204011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 4t8
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1118044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,131044 + 3t + 2t2 + t3
ϕ4,618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,818044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1614044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 11t8 + 10t9 + 9t10 + 8t11 + 7t12 + 6t13 + 5t14
ϕ5,1217055 + 10t + 11t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 8t14 + 4t15
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0
ϕ1,20 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0
ϕ1,40 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0
ϕ2,21 2 0 2 2 0 2 0 2 0 0 0 1 0 1 0 1 2 0 1 0 1 1 2 2 0 0 0
ϕ2,27 2 0 2 0 1 2 0 0 1 0 0 0 0 0 0 1 0 0 1 2 1 1 2 2 0 0 1
ϕ2,11 2 0 2 2 0 2 0 2 0 1 0 0 0 1 0 1 1 1 1 1 0 1 2 2 0 0 0
ϕ2,17 2 0 2 0 1 0 1 2 0 0 0 0 0 0 0 1 1 0 1 1 2 0 2 2 1 0 0
ϕ2,1 2 1 2 2 0 2 0 2 0 1 0 1 0 0 0 2 1 0 0 1 0 1 2 2 0 0 0
ϕ2,7 2 1 2 2 0 0 1 0 1 0 0 0 0 0 0 0 1 0 2 1 0 1 2 2 0 1 0
ϕ3,2 3 0 3 3 0 3 0 1 1 1 0 1 0 1 0 2 2 0 1 1 1 2 3 3 0 0 0
ϕ3,14 3 1 3 1 1 1 1 3 0 0 1 0 0 0 0 2 2 0 1 1 2 0 3 3 0 0 0
ϕ3,10'' 3 0 3 1 1 3 0 3 0 1 0 1 0 1 0 2 2 0 1 1 1 2 3 3 0 0 0
ϕ3,10' 3 0 3 3 0 1 1 1 1 0 0 0 1 0 0 0 2 0 3 1 1 2 3 3 0 0 0
ϕ3,12 3 1 3 3 0 1 1 3 0 1 0 1 0 1 0 2 2 1 1 1 0 1 3 3 0 0 0
ϕ3,6 3 0 3 1 1 3 0 1 1 0 0 0 0 0 1 2 0 0 1 3 1 2 3 3 0 0 0
ϕ4,3 4 0 4 2 1 4 0 2 1 0 0 1 0 1 0 3 2 0 1 2 1 3 4 4 0 0 1
ϕ4,11 4 1 4 2 1 2 1 4 0 1 0 0 0 1 0 3 3 0 1 1 1 2 4 4 1 0 0
ϕ4,13 4 0 4 4 0 2 1 2 1 1 0 1 0 0 0 2 3 1 2 1 1 2 4 4 0 1 0
ϕ4,6 4 1 4 2 1 2 1 2 1 0 0 0 0 0 0 1 2 0 3 2 2 1 4 4 1 1 0
ϕ4,8 4 1 4 2 1 2 1 2 1 0 0 0 0 0 0 1 1 0 3 3 1 2 4 4 0 1 1
ϕ4,16 4 0 4 2 1 2 1 2 1 0 0 0 0 0 0 2 1 0 2 3 3 1 4 4 1 0 1
ϕ5,12 5 0 5 3 1 3 1 3 1 1 0 0 0 0 0 3 2 0 2 3 2 3 5 5 1 1 0
ϕ5,8 5 0 5 3 1 3 1 3 1 0 0 1 0 0 0 3 3 0 2 2 3 2 5 5 0 1 1
ϕ5,4 5 1 5 3 1 3 1 3 1 0 0 0 0 1 0 2 3 0 3 2 1 3 5 5 1 0 1
ϕ6,7 6 1 6 2 2 4 1 4 1 0 0 0 0 0 0 4 2 0 2 4 3 2 6 6 1 1 1
ϕ6,9 6 1 6 4 1 2 2 4 1 0 0 0 0 0 0 2 4 0 4 2 3 2 6 6 1 1 1
ϕ6,5 6 0 6 4 1 4 1 2 2 0 0 0 0 0 0 2 2 0 4 4 2 4 6 6 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 0 t40 t21 0 t11 0 t 0 t2 0 0 0 0 0 t3 0 0 0 t8 0 t12 t8 t4 0 0 0
ϕ1,20 t40 1 t20 t 0 t21 0 t11 0 0 0 t2 0 0 0 t13 t3 0 0 0 0 0 t12 t8 0 0 0
ϕ1,40 t20 0 1 t11 0 t 0 t21 0 0 0 0 0 t2 0 0 t13 t3 t8 0 0 0 t4 t12 0 0 0
ϕ2,21 t21 + t39 0 t + t19 1 + t12 0 t2 + t20 0 t10 + t22 0 0 0 t 0 t3 0 t12 t2 + t14 0 t9 0 t7 t3 t5 + t11 t7 + t13 0 0 0
ϕ2,27 t27 + t33 0 t7 + t13 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 t6 0 0 t15 t5 + t11 t t9 t5 + t11 t7 + t13 0 0 t2
ϕ2,11 t + t19 0 t11 + t29 t10 + t22 0 1 + t12 0 t2 + t20 0 t3 0 0 0 t 0 t4 t12 t2 t7 t9 0 t13 t3 + t9 t5 + t11 0 0 0
ϕ2,17 t7 + t13 0 t17 + t23 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 t10 t6 0 t t15 t5 + t11 0 t3 + t9 t5 + t11 t2 0 0
ϕ2,1 t11 + t29 t t21 + t39 t2 + t20 0 t10 + t22 0 1 + t12 0 t 0 t3 0 0 0 t2 + t14 t4 0 0 t7 0 t11 t7 + t13 t3 + t9 0 0 0
ϕ2,7 t17 + t23 t7 t27 + t33 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 t10 0 t5 + t11 t 0 t5 t7 + t13 t3 + t9 0 t2 0
ϕ3,2 t10 + t22 + t28 0 t2 + t20 + t38 t + t13 + t19 0 t3 + t9 + t21 0 t11 t5 1 0 t2 0 t4 0 t + t13 t3 + t15 0 t10 t6 t8 t4 + t10 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,14 t10 + t16 + t34 t6 t14 + t20 + t26 t7 t7 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 t7 + t13 t3 + t9 0 t4 t12 t2 + t8 0 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,10'' t2 + t20 + t38 0 t12 + t18 + t30 t11 t5 t + t13 + t19 0 t3 + t9 + t21 0 t4 0 1 0 t2 0 t5 + t11 t + t13 0 t8 t10 t6 t2 + t14 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,10' t14 + t20 + t26 0 t6 + t24 + t30 t5 + t11 + t17 0 t7 t7 t15 t3 0 0 0 1 0 0 0 t7 + t13 0 t2 + t8 + t14 t4 t12 t2 + t8 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,12 t12 + t18 + t30 t2 t10 + t22 + t28 t3 + t9 + t21 0 t11 t5 t + t13 + t19 0 t2 0 t4 0 1 0 t3 + t15 t5 + t11 t t6 t8 0 t12 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ3,6 t6 + t24 + t30 0 t10 + t16 + t34 t15 t3 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 t3 + t9 0 0 t12 t2 + t8 + t14 t4 t6 + t12 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 0 t13 + t19 + t31 + t37 t12 + t18 t6 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 t 0 t3 0 1 + t6 + t12 t2 + t14 0 t9 t5 + t11 t7 t3 + t9 + t15 2t5 + 2t11 t + 2t7 + t13 0 0 t2
ϕ4,11 t13 + t19 + t31 + t37 t3 t11 + t17 + t23 + t29 t4 + t10 t4 t12 + t18 t6 t2 + t8 + t14 + t20 0 t3 0 0 0 t 0 t4 + t10 + t16 1 + t6 + t12 0 t7 t9 t5 t + t13 t3 + 2t9 + t15 2t5 + 2t11 t2 0 0
ϕ4,13 t11 + t17 + t23 + t29 0 t3 + t9 + t21 + t27 t2 + t8 + t14 + t20 0 t4 + t10 t4 t12 + t18 t6 t 0 t3 0 0 0 t2 + t14 t4 + t10 + t16 1 t5 + t11 t7 t9 t5 + t11 t + 2t7 + t13 t3 + 2t9 + t15 0 t2 0
ϕ4,6 t6 + t12 + t18 + t24 t8 t16 + t22 + t28 + t34 t9 + t15 t9 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 t9 t5 + t11 0 1 + t6 + t12 t2 + t14 t4 + t10 t6 t2 + 2t8 + t14 2t4 + 2t10 t t3 0
ϕ4,8 t16 + t22 + t28 + t34 t6 t8 + t14 + t26 + t32 t7 + t13 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 t7 t9 0 t4 + t10 + t16 1 + t6 + t12 t2 t4 + t10 2t6 + 2t12 t2 + 2t8 + t14 0 t t3
ϕ4,16 t8 + t14 + t26 + t32 0 t6 + t12 + t18 + t24 t5 + t17 t5 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 t5 + t11 t7 0 t2 + t14 t4 + t10 + t16 1 + t6 + t12 t8 2t4 + 2t10 2t6 + 2t12 t3 0 t
ϕ5,12 t12 + t18 + t24 + t30 + t36 0 t4 + t10 + t16 + t22 + t28 t3 + t9 + t15 t3 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t2 0 0 0 0 0 t3 + t9 + t15 t5 + t11 0 t6 + t12 t2 + t8 + t14 t4 + t10 1 + t6 + t12 t2 + 3t8 + t14 2t4 + 2t10 + t16 t t3 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 0 t8 + t14 + t20 + t26 + t32 t7 + t13 + t19 t7 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 t2 0 0 0 t + t7 + t13 t3 + t9 + t15 0 t4 + t10 t6 + t12 t2 + t8 + t14 t4 + t10 1 + 2t6 + 2t12 t2 + 3t8 + t14 0 t t3
ϕ5,4 t8 + t14 + t20 + t26 + t32 t4 t12 + t18 + t24 + t30 + t36 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 t5 + t11 t + t7 + t13 0 t2 + t8 + t14 t4 + t10 t6 t2 + t8 + t14 2t4 + 2t10 + t16 1 + 2t6 + 2t12 t3 0 t
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 t7 t9 + 2t15 + t21 + t27 + t33 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 0 t5 + t11 t + t7 + 2t13 2t3 + t9 t5 + t11 t + 3t7 + 2t13 2t3 + 3t9 + t15 1 t2 t4
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 t5 t7 + t13 + t19 + 2t25 + t31 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 0 2t3 + t9 + t15 t5 + t11 t + t7 + t13 t3 + t9 3t5 + 3t11 t + 3t7 + 2t13 t4 1 t2
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 0 t5 + t11 + t17 + t23 + t29 + t35 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 t4 + t10 t6 + t12 0 t + t7 + 2t13 2t3 + t9 + t15 t5 + t11 t + 2t7 + t13 2t3 + 3t9 + t15 3t5 + 3t11 t2 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ4,13,  ϕ1,40,  ϕ4,16,  ϕ5,4,  ϕ2,1,  ϕ6,7,  ϕ2,7,  ϕ6,9,  ϕ3,2,  ϕ6,5,  ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ3,6},   {ϕ2,11,  ϕ2,17},   {ϕ2,21,  ϕ2,27},   {ϕ4,11,  ϕ4,8},   {ϕ4,3,  ϕ4,6}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,40311 + 2t
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1322 + t
ϕ2,7642 + 4t
ϕ3,2363
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,121263 + 6t + 3t2
ϕ3,6393
ϕ4,318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1118044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,139124 + 5t
ϕ4,618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,818044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,16684 + 2t
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,49155 + 4t
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,921246 + 9t + 6t2
ϕ6,56246

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0
ϕ1,20 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0
ϕ1,40 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0
ϕ2,21 2 2 1 2 0 2 0 0 0 0 0 1 0 0 0 1 2 0 1 0 0 2 2 0 0 0 0
ϕ2,27 2 2 0 0 1 2 0 0 0 0 0 0 0 0 0 1 0 0 1 2 1 2 2 0 0 0 0
ϕ2,11 2 2 0 2 0 2 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 2 2 0 0 0 0
ϕ2,17 2 2 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 2 2 0 1 0 0
ϕ2,1 2 2 0 2 0 2 0 1 0 0 0 1 0 0 0 2 1 0 0 1 0 2 2 0 0 0 0
ϕ2,7 2 2 0 2 0 0 1 0 1 0 0 0 0 0 0 0 1 0 2 1 0 2 2 0 0 0 0
ϕ3,2 3 3 0 3 0 3 0 0 0 1 0 1 0 0 0 2 2 0 1 1 0 3 3 0 0 0 0
ϕ3,14 3 3 0 1 1 1 1 0 0 0 1 0 0 0 0 2 2 0 1 1 0 3 3 0 0 0 0
ϕ3,10'' 3 3 0 1 1 3 0 0 0 0 0 1 0 1 0 2 2 0 1 1 0 3 3 0 0 0 0
ϕ3,10' 3 3 0 3 0 1 1 0 0 0 0 0 1 0 0 0 2 0 3 1 0 3 3 0 0 0 0
ϕ3,12 3 3 0 3 0 1 1 0 0 0 0 1 0 1 0 2 2 0 1 1 0 3 3 0 0 0 0
ϕ3,6 3 3 0 1 1 3 0 0 0 0 0 0 0 0 1 2 0 0 1 3 0 3 3 0 0 0 0
ϕ4,3 4 4 0 2 1 4 0 0 0 0 0 1 0 0 0 3 2 0 1 2 0 4 4 1 0 0 0
ϕ4,11 4 4 0 2 1 2 1 0 0 0 0 0 0 1 0 3 3 0 1 1 0 4 4 0 1 0 0
ϕ4,13 4 4 0 4 0 2 1 0 0 0 0 1 0 0 0 2 3 1 2 1 0 4 4 0 0 0 0
ϕ4,6 4 4 0 2 1 2 1 0 1 0 0 0 0 0 0 1 2 0 3 2 0 4 4 0 1 0 0
ϕ4,8 4 4 0 2 1 2 1 0 0 0 0 0 0 0 0 1 1 0 3 3 0 4 4 0 0 1 0
ϕ4,16 4 4 0 2 1 2 1 0 0 0 0 0 0 0 0 2 1 0 2 3 1 4 4 0 1 0 0
ϕ5,12 5 5 0 3 1 3 1 0 0 0 0 0 0 0 0 3 2 1 2 3 0 5 5 0 1 0 0
ϕ5,8 5 5 0 3 1 3 1 0 0 0 0 1 0 0 0 3 3 0 2 2 0 5 5 0 0 1 0
ϕ5,4 5 5 0 3 1 3 1 0 0 0 0 0 0 0 0 2 3 0 3 2 0 5 5 1 1 0 0
ϕ6,7 6 6 0 2 2 4 1 0 0 0 0 0 0 0 0 4 2 0 2 4 0 6 6 0 1 1 0
ϕ6,9 6 6 0 4 1 2 2 0 0 0 0 0 0 0 0 2 4 0 4 2 0 6 6 0 1 1 0
ϕ6,5 6 6 0 4 1 4 1 0 0 0 0 0 0 0 0 2 2 0 4 4 0 6 6 0 1 0 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 0 t21 0 t11 0 t 0 0 0 0 0 0 0 t3 0 0 0 t8 0 t12 t8 0 0 0 0
ϕ1,20 t40 1 0 t 0 t21 0 0 0 0 0 t2 0 0 0 t13 t3 0 0 0 0 t4 t12 0 0 0 0
ϕ1,40 t20 t40 1 t11 0 t 0 0 0 0 0 0 0 0 0 0 t13 0 t8 0 0 t8 t4 0 0 0 0
ϕ2,21 t21 + t39 t11 + t29 t 1 + t12 0 t2 + t20 0 0 0 0 0 t 0 0 0 t12 t2 + t14 0 t9 0 0 t3 + t9 t5 + t11 0 0 0 0
ϕ2,27 t27 + t33 t17 + t23 0 0 1 t8 + t14 0 0 0 0 0 0 0 0 0 t6 0 0 t15 t5 + t11 t t3 + t9 t5 + t11 0 0 0 0
ϕ2,11 t + t19 t21 + t39 0 t10 + t22 0 1 + t12 0 0 0 0 0 0 0 t 0 t4 t12 0 t7 t9 0 t7 + t13 t3 + t9 0 0 0 0
ϕ2,17 t7 + t13 t27 + t33 0 0 t10 0 1 0 0 0 0 0 0 0 0 t10 t6 0 t t15 0 t7 + t13 t3 + t9 0 t2 0 0
ϕ2,1 t11 + t29 t + t19 0 t2 + t20 0 t10 + t22 0 1 0 0 0 t3 0 0 0 t2 + t14 t4 0 0 t7 0 t5 + t11 t7 + t13 0 0 0 0
ϕ2,7 t17 + t23 t7 + t13 0 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 t10 0 t5 + t11 t 0 t5 + t11 t7 + t13 0 0 0 0
ϕ3,2 t10 + t22 + t28 t12 + t18 + t30 0 t + t13 + t19 0 t3 + t9 + t21 0 0 0 1 0 t2 0 0 0 t + t13 t3 + t15 0 t10 t6 0 t4 + 2t10 2t6 + t12 0 0 0 0
ϕ3,14 t10 + t16 + t34 t6 + t24 + t30 0 t7 t7 t15 t3 0 0 0 1 0 0 0 0 t7 + t13 t3 + t9 0 t4 t12 0 t4 + 2t10 2t6 + t12 0 0 0 0
ϕ3,10'' t2 + t20 + t38 t10 + t22 + t28 0 t11 t5 t + t13 + t19 0 0 0 0 0 1 0 t2 0 t5 + t11 t + t13 0 t8 t10 0 t2 + t8 + t14 t4 + 2t10 0 0 0 0
ϕ3,10' t14 + t20 + t26 t10 + t16 + t34 0 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 0 0 0 t7 + t13 0 t2 + t8 + t14 t4 0 t2 + t8 + t14 t4 + 2t10 0 0 0 0
ϕ3,12 t12 + t18 + t30 t2 + t20 + t38 0 t3 + t9 + t21 0 t11 t5 0 0 0 0 t4 0 1 0 t3 + t15 t5 + t11 0 t6 t8 0 2t6 + t12 t2 + t8 + t14 0 0 0 0
ϕ3,6 t6 + t24 + t30 t14 + t20 + t26 0 t15 t3 t5 + t11 + t17 0 0 0 0 0 0 0 0 1 t3 + t9 0 0 t12 t2 + t8 + t14 0 2t6 + t12 t2 + t8 + t14 0 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 0 t12 + t18 t6 t2 + t8 + t14 + t20 0 0 0 0 0 t 0 0 0 1 + t6 + t12 t2 + t14 0 t9 t5 + t11 0 t3 + 2t9 + t15 2t5 + 2t11 t 0 0 0
ϕ4,11 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 0 t4 + t10 t4 t12 + t18 t6 0 0 0 0 0 0 t 0 t4 + t10 + t16 1 + t6 + t12 0 t7 t9 0 t + 2t7 + t13 t3 + 2t9 + t15 0 t2 0 0
ϕ4,13 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 0 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 0 0 t3 0 0 0 t2 + t14 t4 + t10 + t16 1 t5 + t11 t7 0 2t5 + 2t11 t + 2t7 + t13 0 0 0 0
ϕ4,6 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 0 t9 + t15 t9 t5 + t17 t5 0 t 0 0 0 0 0 0 t9 t5 + t11 0 1 + t6 + t12 t2 + t14 0 2t6 + 2t12 t2 + 2t8 + t14 0 t 0 0
ϕ4,8 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 0 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 0 0 t7 t9 0 t4 + t10 + t16 1 + t6 + t12 0 2t4 + 2t10 2t6 + 2t12 0 0 t 0
ϕ4,16 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 0 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 0 0 t5 + t11 t7 0 t2 + t14 t4 + t10 + t16 1 t2 + 2t8 + t14 2t4 + 2t10 0 t3 0 0
ϕ5,12 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 0 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 0 0 0 0 0 0 t3 + t9 + t15 t5 + t11 t t6 + t12 t2 + t8 + t14 0 1 + 2t6 + 2t12 t2 + 3t8 + t14 0 t 0 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 0 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 0 0 t + t7 + t13 t3 + t9 + t15 0 t4 + t10 t6 + t12 0 2t4 + 2t10 + t16 1 + 2t6 + 2t12 0 0 t 0
ϕ5,4 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 0 t5 + t11 + t17 t5 t7 + t13 + t19 t7 0 0 0 0 0 0 0 0 t5 + t11 t + t7 + t13 0 t2 + t8 + t14 t4 + t10 0 t2 + 3t8 + t14 2t4 + 2t10 + t16 1 t3 0 0
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 0 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 0 t5 + t11 t + t7 + 2t13 0 3t5 + 3t11 t + 3t7 + 2t13 0 1 t2 0
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 0 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 0 2t3 + t9 + t15 t5 + t11 0 2t3 + 3t9 + t15 3t5 + 3t11 0 t4 1 0
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 0 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 0 0 t4 + t10 t6 + t12 0 t + t7 + 2t13 2t3 + t9 + t15 0 t + 3t7 + 2t13 2t3 + 3t9 + t15 0 t2 0 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ5,8,  ϕ4,13,  ϕ4,16},   {ϕ2,11,  ϕ2,17},   {ϕ2,21,  ϕ2,27},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2},   {ϕ4,11,  ϕ4,8},   {ϕ4,3,  ϕ4,6},   {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)
ϕ1,01011 + 2t + 3t2 + 4t3
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,4036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1118044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1317044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 11t13 + 10t14 + 5t15
ϕ4,618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,818044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,164044 + 8t + 7t2 + 6t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,814055 + 6t + 7t2 + 8t3 + 9t4 + 10t5 + 11t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 8t13 + 4t14
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0
ϕ1,20 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 1 0 0 0
ϕ1,40 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0
ϕ2,21 0 2 2 2 0 2 0 2 0 0 0 1 0 1 0 1 2 1 1 0 1 2 0 2 0 0 0
ϕ2,27 0 2 2 0 1 2 0 0 1 0 0 0 0 0 0 1 0 1 1 2 1 2 0 2 0 0 1
ϕ2,11 1 2 2 2 0 2 0 2 0 1 0 0 0 1 0 1 1 1 1 1 0 2 0 2 0 0 0
ϕ2,17 0 2 2 0 1 0 1 2 0 0 0 0 0 0 0 1 1 0 1 1 0 2 2 2 1 0 0
ϕ2,1 0 2 2 2 0 2 0 2 0 1 0 1 0 0 0 2 1 1 0 1 0 2 1 2 0 0 0
ϕ2,7 0 2 2 2 0 0 1 0 1 0 0 0 0 0 0 0 1 1 2 1 0 2 1 2 0 1 0
ϕ3,2 0 3 3 3 0 3 0 1 1 1 0 1 0 1 0 2 2 2 1 1 0 3 1 3 0 0 0
ϕ3,14 0 3 3 1 1 1 1 3 0 0 1 0 0 0 0 2 2 0 1 1 1 3 2 3 0 0 0
ϕ3,10'' 1 3 3 1 1 3 0 3 0 1 0 1 0 1 0 2 2 1 1 1 1 3 0 3 0 0 0
ϕ3,10' 0 3 3 3 0 1 1 1 1 0 0 0 1 0 0 0 2 2 3 1 0 3 1 3 0 0 0
ϕ3,12 0 3 3 3 0 1 1 3 0 1 0 1 0 1 0 2 2 2 1 1 0 3 1 3 0 0 0
ϕ3,6 0 3 3 1 1 3 0 1 1 0 0 0 0 0 1 2 0 2 1 3 0 3 1 3 0 0 0
ϕ4,3 1 4 4 2 1 4 0 2 1 0 0 1 0 1 0 3 2 2 1 2 0 4 1 4 0 0 1
ϕ4,11 0 4 4 2 1 2 1 4 0 1 0 0 0 1 0 3 3 2 1 1 1 4 1 4 1 0 0
ϕ4,13 0 4 4 4 0 2 1 2 1 1 0 1 0 0 0 2 3 3 2 1 0 4 1 4 0 1 0
ϕ4,6 0 4 4 2 1 2 1 2 1 0 0 0 0 0 0 1 2 1 3 2 0 4 3 4 1 1 0
ϕ4,8 0 4 4 2 1 2 1 2 1 0 0 0 0 0 0 1 1 2 3 3 1 4 1 4 0 1 1
ϕ4,16 0 4 4 2 1 2 1 2 1 0 0 0 0 0 0 2 1 1 2 3 1 4 2 4 1 0 1
ϕ5,12 0 5 5 3 1 3 1 3 1 1 0 0 0 0 0 3 2 3 2 3 1 5 1 5 1 1 0
ϕ5,8 0 5 5 3 1 3 1 3 1 0 0 1 0 0 0 3 3 2 2 2 0 5 3 5 0 1 1
ϕ5,4 0 5 5 3 1 3 1 3 1 0 0 0 0 1 0 2 3 3 3 2 0 5 2 5 1 0 1
ϕ6,7 0 6 6 2 2 4 1 4 1 0 0 0 0 0 0 4 2 2 2 4 1 6 3 6 1 1 1
ϕ6,9 0 6 6 4 1 2 2 4 1 0 0 0 0 0 0 2 4 2 4 2 1 6 3 6 1 1 1
ϕ6,5 0 6 6 4 1 4 1 2 2 0 0 0 0 0 0 2 2 4 4 4 0 6 2 6 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 t40 t21 0 t11 0 t 0 t2 0 0 0 0 0 t3 0 0 0 t8 0 t12 0 t4 0 0 0
ϕ1,20 0 1 t20 t 0 t21 0 t11 0 0 0 t2 0 0 0 t13 t3 0 0 0 t8 t4 0 t8 0 0 0
ϕ1,40 0 t40 1 t11 0 t 0 t21 0 0 0 0 0 t2 0 0 t13 t3 t8 0 0 t8 0 t12 0 0 0
ϕ2,21 0 t11 + t29 t + t19 1 + t12 0 t2 + t20 0 t10 + t22 0 0 0 t 0 t3 0 t12 t2 + t14 t4 t9 0 t7 t3 + t9 0 t7 + t13 0 0 0
ϕ2,27 0 t17 + t23 t7 + t13 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 t6 0 t10 t15 t5 + t11 t t3 + t9 0 t7 + t13 0 0 t2
ϕ2,11 t t21 + t39 t11 + t29 t10 + t22 0 1 + t12 0 t2 + t20 0 t3 0 0 0 t 0 t4 t12 t2 t7 t9 0 t7 + t13 0 t5 + t11 0 0 0
ϕ2,17 0 t27 + t33 t17 + t23 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 t10 t6 0 t t15 0 t7 + t13 t3 + t9 t5 + t11 t2 0 0
ϕ2,1 0 t + t19 t21 + t39 t2 + t20 0 t10 + t22 0 1 + t12 0 t 0 t3 0 0 0 t2 + t14 t4 t12 0 t7 0 t5 + t11 t7 t3 + t9 0 0 0
ϕ2,7 0 t7 + t13 t27 + t33 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 t10 t6 t5 + t11 t 0 t5 + t11 t13 t3 + t9 0 t2 0
ϕ3,2 0 t12 + t18 + t30 t2 + t20 + t38 t + t13 + t19 0 t3 + t9 + t21 0 t11 t5 1 0 t2 0 t4 0 t + t13 t3 + t15 t5 + t11 t10 t6 0 t4 + 2t10 t6 t2 + t8 + t14 0 0 0
ϕ3,14 0 t6 + t24 + t30 t14 + t20 + t26 t7 t7 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 t7 + t13 t3 + t9 0 t4 t12 t2 t4 + 2t10 t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,10'' t2 t10 + t22 + t28 t12 + t18 + t30 t11 t5 t + t13 + t19 0 t3 + t9 + t21 0 t4 0 1 0 t2 0 t5 + t11 t + t13 t3 t8 t10 t6 t2 + t8 + t14 0 2t6 + t12 0 0 0
ϕ3,10' 0 t10 + t16 + t34 t6 + t24 + t30 t5 + t11 + t17 0 t7 t7 t15 t3 0 0 0 1 0 0 0 t7 + t13 t3 + t9 t2 + t8 + t14 t4 0 t2 + t8 + t14 t10 2t6 + t12 0 0 0
ϕ3,12 0 t2 + t20 + t38 t10 + t22 + t28 t3 + t9 + t21 0 t11 t5 t + t13 + t19 0 t2 0 t4 0 1 0 t3 + t15 t5 + t11 t + t13 t6 t8 0 2t6 + t12 t8 t4 + 2t10 0 0 0
ϕ3,6 0 t14 + t20 + t26 t10 + t16 + t34 t15 t3 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 t3 + t9 0 t7 + t13 t12 t2 + t8 + t14 0 2t6 + t12 t2 t4 + 2t10 0 0 0
ϕ4,3 t3 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 t12 + t18 t6 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 t 0 t3 0 1 + t6 + t12 t2 + t14 t4 + t10 t9 t5 + t11 0 t3 + 2t9 + t15 t5 t + 2t7 + t13 0 0 t2
ϕ4,11 0 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 t4 + t10 t4 t12 + t18 t6 t2 + t8 + t14 + t20 0 t3 0 0 0 t 0 t4 + t10 + t16 1 + t6 + t12 t2 + t14 t7 t9 t5 t + 2t7 + t13 t9 2t5 + 2t11 t2 0 0
ϕ4,13 0 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 t2 + t8 + t14 + t20 0 t4 + t10 t4 t12 + t18 t6 t 0 t3 0 0 0 t2 + t14 t4 + t10 + t16 1 + t6 + t12 t5 + t11 t7 0 2t5 + 2t11 t7 t3 + 2t9 + t15 0 t2 0
ϕ4,6 0 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 t9 + t15 t9 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 t9 t5 + t11 t7 1 + t6 + t12 t2 + t14 0 2t6 + 2t12 t2 + t8 + t14 2t4 + 2t10 t t3 0
ϕ4,8 0 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 t7 + t13 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 t7 t9 t5 + t11 t4 + t10 + t16 1 + t6 + t12 t2 2t4 + 2t10 t12 t2 + 2t8 + t14 0 t t3
ϕ4,16 0 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 t5 + t17 t5 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 t5 + t11 t7 t9 t2 + t14 t4 + t10 + t16 1 t2 + 2t8 + t14 t4 + t10 2t6 + 2t12 t3 0 t
ϕ5,12 0 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 t3 + t9 + t15 t3 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t2 0 0 0 0 0 t3 + t9 + t15 t5 + t11 t + t7 + t13 t6 + t12 t2 + t8 + t14 t4 1 + 2t6 + 2t12 t8 2t4 + 2t10 + t16 t t3 0
ϕ5,8 0 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 t7 + t13 + t19 t7 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 t2 0 0 0 t + t7 + t13 t3 + t9 + t15 t5 + t11 t4 + t10 t6 + t12 0 2t4 + 2t10 + t16 1 + t6 + t12 t2 + 3t8 + t14 0 t t3
ϕ5,4 0 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 t5 + t11 t + t7 + t13 t3 + t9 + t15 t2 + t8 + t14 t4 + t10 0 t2 + 3t8 + t14 t4 + t10 1 + 2t6 + 2t12 t3 0 t
ϕ6,7 0 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 t6 + t12 t5 + t11 t + t7 + 2t13 t3 3t5 + 3t11 t + t7 + t13 2t3 + 3t9 + t15 1 t2 t4
ϕ6,9 0 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 t4 + t10 2t3 + t9 + t15 t5 + t11 t 2t3 + 3t9 + t15 t5 + 2t11 t + 3t7 + 2t13 t4 1 t2
ϕ6,5 0 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 t4 + t10 t6 + t12 t2 + 2t8 + t14 t + t7 + 2t13 2t3 + t9 + t15 0 t + 3t7 + 2t13 t3 + t9 3t5 + 3t11 t2 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,11,  ϕ2,17},   {ϕ2,21,  ϕ2,27},   {ϕ4,3,  ϕ4,6},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2},   {ϕ5,12,  ϕ1,40,  ϕ4,11,  ϕ4,8},   {ϕ4,13,  ϕ4,16},   {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)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,401011 + 2t + 3t2 + 4t3
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1117044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 11t13 + 10t14 + 5t15
ϕ4,1318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,84044 + 8t + 7t2 + 6t3 + 5t4 + 4t5 + 3t6 + 2t7 + t8
ϕ4,1618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ5,1214055 + 6t + 7t2 + 8t3 + 9t4 + 10t5 + 11t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 8t13 + 4t14
ϕ5,836055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0
ϕ1,20 1 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0
ϕ1,40 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 0
ϕ2,21 2 2 1 2 0 2 0 2 0 0 0 1 0 1 0 1 1 1 1 0 1 0 2 2 0 0 0
ϕ2,27 2 2 0 0 1 2 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 2 2 2 0 0 1
ϕ2,11 2 2 0 2 0 2 0 2 0 1 0 0 0 1 0 1 1 2 1 0 0 1 2 2 0 0 0
ϕ2,17 2 2 0 0 1 0 1 2 0 0 0 0 0 0 0 1 1 0 1 0 2 1 2 2 1 0 0
ϕ2,1 2 2 0 2 0 2 0 2 0 1 0 1 0 0 0 2 1 1 0 1 1 0 2 2 0 0 0
ϕ2,7 2 2 0 2 0 0 1 0 1 0 0 0 0 0 0 0 1 1 2 1 1 0 2 2 0 1 0
ϕ3,2 3 3 1 3 0 3 0 1 1 1 0 1 0 1 0 2 1 2 1 1 1 0 3 3 0 0 0
ϕ3,14 3 3 0 1 1 1 1 3 0 0 1 0 0 0 0 2 2 0 1 0 3 1 3 3 0 0 0
ϕ3,10'' 3 3 0 1 1 3 0 3 0 1 0 1 0 1 0 2 2 2 1 0 1 1 3 3 0 0 0
ϕ3,10' 3 3 0 3 0 1 1 1 1 0 0 0 1 0 0 0 2 2 3 0 1 1 3 3 0 0 0
ϕ3,12 3 3 0 3 0 1 1 3 0 1 0 1 0 1 0 2 2 2 1 0 1 1 3 3 0 0 0
ϕ3,6 3 3 0 1 1 3 0 1 1 0 0 0 0 0 1 2 0 2 1 1 1 2 3 3 0 0 0
ϕ4,3 4 4 0 2 1 4 0 2 1 0 0 1 0 1 0 3 2 3 1 1 1 1 4 4 0 0 1
ϕ4,11 4 4 0 2 1 2 1 4 0 1 0 0 0 1 0 3 3 2 1 0 2 1 4 4 1 0 0
ϕ4,13 4 4 1 4 0 2 1 2 1 1 0 1 0 0 0 2 2 3 2 0 1 1 4 4 0 1 0
ϕ4,6 4 4 0 2 1 2 1 2 1 0 0 0 0 0 0 1 2 1 3 1 3 1 4 4 1 1 0
ϕ4,8 4 4 0 2 1 2 1 2 1 0 0 0 0 0 0 1 1 2 3 1 2 2 4 4 0 1 1
ϕ4,16 4 4 0 2 1 2 1 2 1 0 0 0 0 0 0 2 1 1 2 0 3 3 4 4 1 0 1
ϕ5,12 5 5 0 3 1 3 1 3 1 1 0 0 0 0 0 3 2 3 2 0 2 3 5 5 1 1 0
ϕ5,8 5 5 0 3 1 3 1 3 1 0 0 1 0 0 0 3 3 2 2 0 3 2 5 5 0 1 1
ϕ5,4 5 5 0 3 1 3 1 3 1 0 0 0 0 1 0 2 3 3 3 1 2 1 5 5 1 0 1
ϕ6,7 6 6 0 2 2 4 1 4 1 0 0 0 0 0 0 4 2 2 2 1 4 3 6 6 1 1 1
ϕ6,9 6 6 0 4 1 2 2 4 1 0 0 0 0 0 0 2 4 2 4 0 4 2 6 6 1 1 1
ϕ6,5 6 6 0 4 1 4 1 2 2 0 0 0 0 0 0 2 2 4 4 1 2 3 6 6 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 0 t21 0 t11 0 t 0 t2 0 0 0 0 0 t3 0 t13 0 t8 0 0 t8 t4 0 0 0
ϕ1,20 t40 1 0 t 0 t21 0 t11 0 0 0 t2 0 0 0 t13 t3 0 0 0 t8 0 t12 t8 0 0 0
ϕ1,40 t20 t40 1 t11 0 t 0 t21 0 0 0 0 0 t2 0 0 0 t3 t8 0 0 0 t4 t12 0 0 0
ϕ2,21 t21 + t39 t11 + t29 t 1 + t12 0 t2 + t20 0 t10 + t22 0 0 0 t 0 t3 0 t12 t2 t4 t9 0 t7 0 t5 + t11 t7 + t13 0 0 0
ϕ2,27 t27 + t33 t17 + t23 0 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 t6 0 t10 t15 0 t t3 + t9 t5 + t11 t7 + t13 0 0 t2
ϕ2,11 t + t19 t21 + t39 0 t10 + t22 0 1 + t12 0 t2 + t20 0 t3 0 0 0 t 0 t4 t12 t2 + t14 t7 0 0 t7 t3 + t9 t5 + t11 0 0 0
ϕ2,17 t7 + t13 t27 + t33 0 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 t10 t6 0 t 0 t5 + t11 t13 t3 + t9 t5 + t11 t2 0 0
ϕ2,1 t11 + t29 t + t19 0 t2 + t20 0 t10 + t22 0 1 + t12 0 t 0 t3 0 0 0 t2 + t14 t4 t12 0 t7 t9 0 t7 + t13 t3 + t9 0 0 0
ϕ2,7 t17 + t23 t7 + t13 0 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 t10 t6 t5 + t11 t t15 0 t7 + t13 t3 + t9 0 t2 0
ϕ3,2 t10 + t22 + t28 t12 + t18 + t30 t2 t + t13 + t19 0 t3 + t9 + t21 0 t11 t5 1 0 t2 0 t4 0 t + t13 t3 t5 + t11 t10 t6 t8 0 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,14 t10 + t16 + t34 t6 + t24 + t30 0 t7 t7 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 t7 + t13 t3 + t9 0 t4 0 t2 + t8 + t14 t10 2t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,10'' t2 + t20 + t38 t10 + t22 + t28 0 t11 t5 t + t13 + t19 0 t3 + t9 + t21 0 t4 0 1 0 t2 0 t5 + t11 t + t13 t3 + t15 t8 0 t6 t8 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,10' t14 + t20 + t26 t10 + t16 + t34 0 t5 + t11 + t17 0 t7 t7 t15 t3 0 0 0 1 0 0 0 t7 + t13 t3 + t9 t2 + t8 + t14 0 t12 t2 t4 + 2t10 2t6 + t12 0 0 0
ϕ3,12 t12 + t18 + t30 t2 + t20 + t38 0 t3 + t9 + t21 0 t11 t5 t + t13 + t19 0 t2 0 t4 0 1 0 t3 + t15 t5 + t11 t + t13 t6 0 t10 t6 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ3,6 t6 + t24 + t30 t14 + t20 + t26 0 t15 t3 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 t3 + t9 0 t7 + t13 t12 t2 t4 t6 + t12 t2 + t8 + t14 t4 + 2t10 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 0 t12 + t18 t6 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 t 0 t3 0 1 + t6 + t12 t2 + t14 t4 + t10 + t16 t9 t5 t7 t9 2t5 + 2t11 t + 2t7 + t13 0 0 t2
ϕ4,11 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 0 t4 + t10 t4 t12 + t18 t6 t2 + t8 + t14 + t20 0 t3 0 0 0 t 0 t4 + t10 + t16 1 + t6 + t12 t2 + t14 t7 0 t5 + t11 t7 t3 + 2t9 + t15 2t5 + 2t11 t2 0 0
ϕ4,13 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 t3 t2 + t8 + t14 + t20 0 t4 + t10 t4 t12 + t18 t6 t 0 t3 0 0 0 t2 + t14 t4 + t10 1 + t6 + t12 t5 + t11 0 t9 t5 t + 2t7 + t13 t3 + 2t9 + t15 0 t2 0
ϕ4,6 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 0 t9 + t15 t9 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 t9 t5 + t11 t7 1 + t6 + t12 t2 t4 + t10 + t16 t12 t2 + 2t8 + t14 2t4 + 2t10 t t3 0
ϕ4,8 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 0 t7 + t13 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 t7 t9 t5 + t11 t4 + t10 + t16 1 t2 + t14 t4 + t10 2t6 + 2t12 t2 + 2t8 + t14 0 t t3
ϕ4,16 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 0 t5 + t17 t5 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 t5 + t11 t7 t9 t2 + t14 0 1 + t6 + t12 t2 + t8 + t14 2t4 + 2t10 2t6 + 2t12 t3 0 t
ϕ5,12 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 0 t3 + t9 + t15 t3 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t2 0 0 0 0 0 t3 + t9 + t15 t5 + t11 t + t7 + t13 t6 + t12 0 t4 + t10 1 + t6 + t12 t2 + 3t8 + t14 2t4 + 2t10 + t16 t t3 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 0 t7 + t13 + t19 t7 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 t2 0 0 0 t + t7 + t13 t3 + t9 + t15 t5 + t11 t4 + t10 0 t2 + t8 + t14 t4 + t10 1 + 2t6 + 2t12 t2 + 3t8 + t14 0 t t3
ϕ5,4 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 0 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 t5 + t11 t + t7 + t13 t3 + t9 + t15 t2 + t8 + t14 t4 t6 + t12 t8 2t4 + 2t10 + t16 1 + 2t6 + 2t12 t3 0 t
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 0 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 t6 + t12 t5 + t11 t 2t3 + t9 + t15 t5 + 2t11 t + 3t7 + 2t13 2t3 + 3t9 + t15 1 t2 t4
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 0 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 t4 + t10 2t3 + t9 + t15 0 t + t7 + 2t13 t3 + t9 3t5 + 3t11 t + 3t7 + 2t13 t4 1 t2
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 0 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 t4 + t10 t6 + t12 t2 + 2t8 + t14 t + t7 + 2t13 t3 t5 + t11 t + t7 + t13 2t3 + 3t9 + t15 3t5 + 3t11 t2 t4 1

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ5,8,  ϕ1,40,  ϕ4,3,  ϕ4,6},   {ϕ2,11,  ϕ2,17},   {ϕ2,21,  ϕ2,27},   {ϕ3,14,  ϕ3,10'',  ϕ3,10',  ϕ3,12,  ϕ6,7,  ϕ3,6,  ϕ6,9,  ϕ6,5,  ϕ3,2},   {ϕ4,11,  ϕ4,8},   {ϕ4,13,  ϕ4,16},   {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)
ϕ1,036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,2036011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 9t8 + 10t9 + 11t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 12t18 + 12t19 + 12t20 + 12t21 + 12t22 + 12t23 + 12t24 + 12t25 + 12t26 + 12t27 + 12t28 + 12t29 + 11t30 + 10t31 + 9t32 + 8t33 + 7t34 + 6t35 + 5t36 + 4t37 + 3t38 + 2t39 + t40
ϕ1,404011 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 7t6 + 8t7 + 4t8
ϕ2,2121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,277242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,1121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,177242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,121622 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 12t15 + 12t16 + 12t17 + 10t18 + 8t19 + 6t20 + 4t21 + 2t22
ϕ2,77242 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,22733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,14393
ϕ3,10''2733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,10'393
ϕ3,122733 + 6t + 9t2 + 6t3 + 3t4
ϕ3,6393
ϕ4,31044 + 3t + 2t2 + t3
ϕ4,1118044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1318044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,614044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 11t8 + 10t9 + 9t10 + 8t11 + 7t12 + 6t13 + 5t14
ϕ4,818044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ4,1618044 + 8t + 12t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 12t14 + 8t15 + 4t16
ϕ5,1236055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ5,817055 + 10t + 11t2 + 12t3 + 12t4 + 12t5 + 12t6 + 12t7 + 12t8 + 12t9 + 12t10 + 12t11 + 12t12 + 12t13 + 8t14 + 4t15
ϕ5,436055 + 10t + 15t2 + 20t3 + 25t4 + 30t5 + 30t6 + 30t7 + 30t8 + 30t9 + 30t10 + 30t11 + 25t12 + 20t13 + 15t14 + 10t15 + 5t16
ϕ6,742126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,942126 + 9t + 12t2 + 9t3 + 6t4
ϕ6,542126 + 9t + 12t2 + 9t3 + 6t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0
ϕ1,20 1 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 1 0 0 0
ϕ1,40 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0
ϕ2,21 2 2 1 2 0 2 0 2 0 0 0 1 0 1 0 0 2 1 0 0 1 2 1 2 0 0 0
ϕ2,27 2 2 1 0 1 2 0 0 1 0 0 0 0 0 0 0 0 1 0 2 1 2 1 2 0 0 1
ϕ2,11 2 2 0 2 0 2 0 2 0 1 0 0 0 1 0 0 1 2 1 1 0 2 1 2 0 0 0
ϕ2,17 2 2 0 0 1 0 1 2 0 0 0 0 0 0 0 0 1 0 1 1 2 2 1 2 1 0 0
ϕ2,1 2 2 0 2 0 2 0 2 0 1 0 1 0 0 0 1 1 1 0 1 1 2 1 2 0 0 0
ϕ2,7 2 2 0 2 0 0 1 0 1 0 0 0 0 0 0 0 1 1 2 1 1 2 0 2 0 1 0
ϕ3,2 3 3 1 3 0 3 0 1 1 1 0 1 0 1 0 1 2 2 0 1 1 3 1 3 0 0 0
ϕ3,14 3 3 0 1 1 1 1 3 0 0 1 0 0 0 0 0 2 0 1 1 3 3 2 3 0 0 0
ϕ3,10'' 3 3 0 1 1 3 0 3 0 1 0 1 0 1 0 0 2 2 1 1 1 3 2 3 0 0 0
ϕ3,10' 3 3 1 3 0 1 1 1 1 0 0 0 1 0 0 0 2 2 2 1 1 3 0 3 0 0 0
ϕ3,12 3 3 0 3 0 1 1 3 0 1 0 1 0 1 0 0 2 2 1 1 1 3 2 3 0 0 0
ϕ3,6 3 3 0 1 1 3 0 1 1 0 0 0 0 0 1 0 0 2 1 3 1 3 2 3 0 0 0
ϕ4,3 4 4 0 2 1 4 0 2 1 0 0 1 0 1 0 1 2 3 1 2 1 4 2 4 0 0 1
ϕ4,11 4 4 0 2 1 2 1 4 0 1 0 0 0 1 0 0 3 2 1 1 2 4 3 4 1 0 0
ϕ4,13 4 4 1 4 0 2 1 2 1 1 0 1 0 0 0 0 3 3 1 1 1 4 2 4 0 1 0
ϕ4,6 4 4 0 2 1 2 1 2 1 0 0 0 0 0 0 0 2 1 3 2 3 4 1 4 1 1 0
ϕ4,8 4 4 1 2 1 2 1 2 1 0 0 0 0 0 0 0 1 2 2 3 2 4 1 4 0 1 1
ϕ4,16 4 4 1 2 1 2 1 2 1 0 0 0 0 0 0 0 1 1 1 3 3 4 2 4 1 0 1
ϕ5,12 5 5 1 3 1 3 1 3 1 1 0 0 0 0 0 0 2 3 1 3 2 5 3 5 1 1 0
ϕ5,8 5 5 0 3 1 3 1 3 1 0 0 1 0 0 0 0 3 2 2 2 3 5 3 5 0 1 1
ϕ5,4 5 5 0 3 1 3 1 3 1 0 0 0 0 1 0 0 3 3 3 2 2 5 2 5 1 0 1
ϕ6,7 6 6 0 2 2 4 1 4 1 0 0 0 0 0 0 0 2 2 2 4 4 6 4 6 1 1 1
ϕ6,9 6 6 1 4 1 2 2 4 1 0 0 0 0 0 0 0 4 2 3 2 4 6 2 6 1 1 1
ϕ6,5 6 6 1 4 1 4 1 2 2 0 0 0 0 0 0 0 2 4 3 4 2 6 2 6 1 1 1

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,20) L(ϕ1,40) L(ϕ2,21) L(ϕ2,27) L(ϕ2,11) L(ϕ2,17) L(ϕ2,1) L(ϕ2,7) L(ϕ3,2) L(ϕ3,14) L(ϕ3,10'') L(ϕ3,10') L(ϕ3,12) L(ϕ3,6) L(ϕ4,3) L(ϕ4,11) L(ϕ4,13) L(ϕ4,6) L(ϕ4,8) L(ϕ4,16) L(ϕ5,12) L(ϕ5,8) L(ϕ5,4) L(ϕ6,7) L(ϕ6,9) L(ϕ6,5)
ϕ1,0 1 t20 0 t21 0 t11 0 t 0 t2 0 0 0 0 0 t3 0 t13 0 t8 0 t12 0 t4 0 0 0
ϕ1,20 t40 1 0 t 0 t21 0 t11 0 0 0 t2 0 0 0 0 t3 0 0 0 t8 t4 t12 t8 0 0 0
ϕ1,40 t20 t40 1 t11 0 t 0 t21 0 0 0 0 0 t2 0 0 t13 t3 0 0 0 t8 0 t12 0 0 0
ϕ2,21 t21 + t39 t11 + t29 t 1 + t12 0 t2 + t20 0 t10 + t22 0 0 0 t 0 t3 0 0 t2 + t14 t4 0 0 t7 t3 + t9 t11 t7 + t13 0 0 0
ϕ2,27 t27 + t33 t17 + t23 t7 0 1 t8 + t14 0 0 t10 0 0 0 0 0 0 0 0 t10 0 t5 + t11 t t3 + t9 t5 t7 + t13 0 0 t2
ϕ2,11 t + t19 t21 + t39 0 t10 + t22 0 1 + t12 0 t2 + t20 0 t3 0 0 0 t 0 0 t12 t2 + t14 t7 t9 0 t7 + t13 t3 t5 + t11 0 0 0
ϕ2,17 t7 + t13 t27 + t33 0 0 t10 0 1 t8 + t14 0 0 0 0 0 0 0 0 t6 0 t t15 t5 + t11 t7 + t13 t9 t5 + t11 t2 0 0
ϕ2,1 t11 + t29 t + t19 0 t2 + t20 0 t10 + t22 0 1 + t12 0 t 0 t3 0 0 0 t2 t4 t12 0 t7 t9 t5 + t11 t13 t3 + t9 0 0 0
ϕ2,7 t17 + t23 t7 + t13 0 t8 + t14 0 0 t10 0 1 0 0 0 0 0 0 0 t10 t6 t5 + t11 t t15 t5 + t11 0 t3 + t9 0 t2 0
ϕ3,2 t10 + t22 + t28 t12 + t18 + t30 t2 t + t13 + t19 0 t3 + t9 + t21 0 t11 t5 1 0 t2 0 t4 0 t t3 + t15 t5 + t11 0 t6 t8 t4 + 2t10 t12 t2 + t8 + t14 0 0 0
ϕ3,14 t10 + t16 + t34 t6 + t24 + t30 0 t7 t7 t15 t3 t5 + t11 + t17 0 0 1 0 0 0 0 0 t3 + t9 0 t4 t12 t2 + t8 + t14 t4 + 2t10 t6 + t12 t2 + t8 + t14 0 0 0
ϕ3,10'' t2 + t20 + t38 t10 + t22 + t28 0 t11 t5 t + t13 + t19 0 t3 + t9 + t21 0 t4 0 1 0 t2 0 0 t + t13 t3 + t15 t8 t10 t6 t2 + t8 + t14 t4 + t10 2t6 + t12 0 0 0
ϕ3,10' t14 + t20 + t26 t10 + t16 + t34 t6 t5 + t11 + t17 0 t7 t7 t15 t3 0 0 0 1 0 0 0 t7 + t13 t3 + t9 t2 + t8 t4 t12 t2 + t8 + t14 0 2t6 + t12 0 0 0
ϕ3,12 t12 + t18 + t30 t2 + t20 + t38 0 t3 + t9 + t21 0 t11 t5 t + t13 + t19 0 t2 0 t4 0 1 0 0 t5 + t11 t + t13 t6 t8 t10 2t6 + t12 t2 + t14 t4 + 2t10 0 0 0
ϕ3,6 t6 + t24 + t30 t14 + t20 + t26 0 t15 t3 t5 + t11 + t17 0 t7 t7 0 0 0 0 0 1 0 0 t7 + t13 t12 t2 + t8 + t14 t4 2t6 + t12 t2 + t8 t4 + 2t10 0 0 0
ϕ4,3 t3 + t9 + t21 + t27 t11 + t17 + t23 + t29 0 t12 + t18 t6 t2 + t8 + t14 + t20 0 t4 + t10 t4 0 0 t 0 t3 0 1 t2 + t14 t4 + t10 + t16 t9 t5 + t11 t7 t3 + 2t9 + t15 t5 + t11 t + 2t7 + t13 0 0 t2
ϕ4,11 t13 + t19 + t31 + t37 t3 + t9 + t21 + t27 0 t4 + t10 t4 t12 + t18 t6 t2 + t8 + t14 + t20 0 t3 0 0 0 t 0 0 1 + t6 + t12 t2 + t14 t7 t9 t5 + t11 t + 2t7 + t13 t3 + t9 + t15 2t5 + 2t11 t2 0 0
ϕ4,13 t11 + t17 + t23 + t29 t13 + t19 + t31 + t37 t3 t2 + t8 + t14 + t20 0 t4 + t10 t4 t12 + t18 t6 t 0 t3 0 0 0 0 t4 + t10 + t16 1 + t6 + t12 t5 t7 t9 2t5 + 2t11 t + t13 t3 + 2t9 + t15 0 t2 0
ϕ4,6 t6 + t12 + t18 + t24 t8 + t14 + t26 + t32 0 t9 + t15 t9 t5 + t17 t5 t7 + t13 t 0 0 0 0 0 0 0 t5 + t11 t7 1 + t6 + t12 t2 + t14 t4 + t10 + t16 2t6 + 2t12 t8 2t4 + 2t10 t t3 0
ϕ4,8 t16 + t22 + t28 + t34 t6 + t12 + t18 + t24 t8 t7 + t13 t t9 + t15 t9 t5 + t17 t5 0 0 0 0 0 0 0 t9 t5 + t11 t4 + t10 1 + t6 + t12 t2 + t14 2t4 + 2t10 t6 t2 + 2t8 + t14 0 t t3
ϕ4,16 t8 + t14 + t26 + t32 t16 + t22 + t28 + t34 t6 t5 + t17 t5 t7 + t13 t t9 + t15 t9 0 0 0 0 0 0 0 t7 t9 t2 t4 + t10 + t16 1 + t6 + t12 t2 + 2t8 + t14 t4 + t10 2t6 + 2t12 t3 0 t
ϕ5,12 t12 + t18 + t24 + t30 + t36 t8 + t14 + t20 + t26 + t32 t4 t3 + t9 + t15 t3 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t2 0 0 0 0 0 0 t5 + t11 t + t7 + t13 t6 t2 + t8 + t14 t4 + t10 1 + 2t6 + 2t12 t2 + t8 + t14 2t4 + 2t10 + t16 t t3 0
ϕ5,8 t4 + t10 + t16 + t22 + t28 t12 + t18 + t24 + t30 + t36 0 t7 + t13 + t19 t7 t3 + t9 + t15 t3 t5 + t11 + t17 t5 0 0 t2 0 0 0 0 t3 + t9 + t15 t5 + t11 t4 + t10 t6 + t12 t2 + t8 + t14 2t4 + 2t10 + t16 1 + t6 + t12 t2 + 3t8 + t14 0 t t3
ϕ5,4 t8 + t14 + t20 + t26 + t32 t4 + t10 + t16 + t22 + t28 0 t5 + t11 + t17 t5 t7 + t13 + t19 t7 t3 + t9 + t15 t3 0 0 0 0 t2 0 0 t + t7 + t13 t3 + t9 + t15 t2 + t8 + t14 t4 + t10 t6 + t12 t2 + 3t8 + t14 t4 + t10 1 + 2t6 + 2t12 t3 0 t
ϕ6,7 t5 + t11 + t17 + t23 + t29 + t35 t7 + t13 + t19 + 2t25 + t31 0 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 0 0 0 0 0 0 0 t4 + t10 t6 + t12 t5 + t11 t + t7 + 2t13 2t3 + t9 + t15 3t5 + 3t11 t + 2t7 + t13 2t3 + 3t9 + t15 1 t2 t4
ϕ6,9 t9 + 2t15 + t21 + t27 + t33 t5 + t11 + t17 + t23 + t29 + t35 t7 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 t4 + t10 + 2t16 t4 0 0 0 0 0 0 0 t2 + 2t8 + t14 t4 + t10 2t3 + t9 t5 + t11 t + t7 + 2t13 2t3 + 3t9 + t15 t5 + t11 t + 3t7 + 2t13 t4 1 t2
ϕ6,5 t7 + t13 + t19 + 2t25 + t31 t9 + 2t15 + t21 + t27 + t33 t5 t4 + t10 + 2t16 t4 2t6 + t12 + t18 t6 t8 + t14 t2 + t8 0 0 0 0 0 0 0 t6 + t12 t2 + 2t8 + t14 t + t7 + t13 2t3 + t9 + t15 t5 + t11 t + 3t7 + 2t13 t3 + t9 3t5 + 3t11 t2 t4 1