The representation theory of the restricted rational Cherednik algebra for G5

Computed by Ulrich Thiel using CHAMP (see LMS J. Comput. Math., 2015). Last update on Fri Mar 27 12:48:13 CET 2015.

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For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,4,  ϕ3,2,  ϕ3,6}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,07211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,4'7211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,8'7211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,4''7211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,8''7211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,12'7211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,8'''7211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,12''7211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ1,167211 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 6t8 + 6t9 + 6t10 + 6t11 + 5t12 + 4t13 + 3t14 + 2t15 + t16
ϕ2,97222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,7'7222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,5'7222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,7''7222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,5''7222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,3'7222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,5'''7222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,3''7222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ2,17222 + 4t + 6t2 + 8t3 + 10t4 + 12t5 + 10t6 + 8t7 + 6t8 + 4t9 + 2t10
ϕ3,42433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,62433 + 6t + 6t2 + 6t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4') L(ϕ1,8') L(ϕ1,4'') L(ϕ1,8'') L(ϕ1,12') L(ϕ1,8''') L(ϕ1,12'') L(ϕ1,16) L(ϕ2,9) L(ϕ2,7') L(ϕ2,5') L(ϕ2,7'') L(ϕ2,5'') L(ϕ2,3') L(ϕ2,5''') L(ϕ2,3'') L(ϕ2,1) L(ϕ3,4) L(ϕ3,2) L(ϕ3,6)
ϕ1,0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0
ϕ1,4' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1
ϕ1,8' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
ϕ1,4'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1
ϕ1,8'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
ϕ1,12' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0
ϕ1,8''' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
ϕ1,12'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0
ϕ1,16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1
ϕ2,9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 1
ϕ2,7' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 0
ϕ2,5' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 1 1
ϕ2,7'' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 0
ϕ2,5'' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 1 1
ϕ2,3' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 1
ϕ2,5''' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 1 1
ϕ2,3'' 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 1
ϕ2,1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 0
ϕ3,4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1
ϕ3,2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1
ϕ3,6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1

Graded characters of the simple modules

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

Exceptional hyperplanes

Unknown