The representation theory of the restricted rational Cherednik algebra for G12

Computed by Ulrich Thiel using CHAMP (see LMS J. Comput. Math., 2015). Last update on Fri Mar 27 12:48:16 CET 2015.

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Quick navigation: Exceptional hyperplanes

For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ2,4,  ϕ2,5,  ϕ4,3}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,04811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ1,124811 + 2t + 3t2 + 4t3 + 5t4 + 6t5 + 6t6 + 6t7 + 5t8 + 4t9 + 3t10 + 2t11 + t12
ϕ2,1822 + 4t + 2t2
ϕ2,4242
ϕ2,5822 + 4t + 2t2
ϕ3,24833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ3,64833 + 6t + 6t2 + 6t3 + 6t4 + 6t5 + 6t6 + 6t7 + 3t8
ϕ4,31484 + 6t + 4t2

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,12) L(ϕ2,1) L(ϕ2,4) L(ϕ2,5) L(ϕ3,2) L(ϕ3,6) L(ϕ4,3)
ϕ1,0 1 1 1 0 0 1 1 0
ϕ1,12 1 1 0 0 1 1 1 0
ϕ2,1 2 2 1 0 1 2 2 0
ϕ2,4 2 2 0 1 0 2 2 0
ϕ2,5 2 2 1 0 1 2 2 0
ϕ3,2 3 3 0 0 1 3 3 1
ϕ3,6 3 3 1 0 0 3 3 1
ϕ4,3 4 4 0 0 0 4 4 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,12) L(ϕ2,1) L(ϕ2,4) L(ϕ2,5) L(ϕ3,2) L(ϕ3,6) L(ϕ4,3)
ϕ1,0 1 t12 t 0 0 t2 t6 0
ϕ1,12 t12 1 0 0 t t6 t2 0
ϕ2,1 t5 + t7 t + t11 1 0 t2 t + t7 t3 + t5 0
ϕ2,4 t4 + t8 t4 + t8 0 1 0 t2 + t6 t2 + t6 0
ϕ2,5 t + t11 t5 + t7 t2 0 1 t3 + t5 t + t7 0
ϕ3,2 t2 + t4 + t6 t6 + t8 + t10 0 0 t 1 + t4 + t6 t2 + t4 + t8 t
ϕ3,6 t6 + t8 + t10 t2 + t4 + t6 t 0 0 t2 + t4 + t8 1 + t4 + t6 t
ϕ4,3 t3 + t5 + t7 + t9 t3 + t5 + t7 + t9 0 0 0 t + t3 + t5 + t7 t + t3 + t5 + t7 1 + t2

Exceptional hyperplanes

There are none.