The representation theory of the restricted rational Cherednik algebra for G4

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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Quick navigation: Exceptional hyperplanes

For generic parameters

Non-singleton Calogero–Moser families

There are none.

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,02411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,42411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,82411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ2,52422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,32422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 1 1 1 1 1 1
ϕ1,4 1 1 1 1 1 1 1
ϕ1,8 1 1 1 1 1 1 1
ϕ2,5 2 2 2 2 2 2 2
ϕ2,3 2 2 2 2 2 2 2
ϕ2,1 2 2 2 2 2 2 2
ϕ3,2 3 3 3 3 3 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 t4 t8 t5 t3 t t2
ϕ1,4 t8 1 t4 t t5 t3 t2
ϕ1,8 t4 t8 1 t3 t t5 t2
ϕ2,5 t5 + t7 t3 + t5 t + t3 1 + t4 t2 + t4 t2 + t6 t + t3
ϕ2,3 t + t3 t5 + t7 t3 + t5 t2 + t6 1 + t4 t2 + t4 t + t3
ϕ2,1 t3 + t5 t + t3 t5 + t7 t2 + t4 t2 + t6 1 + t4 t + t3
ϕ3,2 t2 + t4 + t6 t2 + t4 + t6 t2 + t4 + t6 t + t3 + t5 t + t3 + t5 t + t3 + t5 1 + t2 + t4

Exceptional hyperplanes

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

For the generic point of the hyperplane k1,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ1,4,  ϕ2,1},   {ϕ2,5,  ϕ2,3}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0911 + 2t + 3t2 + 2t3 + t4
ϕ1,4111
ϕ1,82411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ2,5822 + 4t + 2t2
ϕ2,31622 + 4t + 4t2 + 4t3 + 2t4
ϕ2,1742 + 3t + 2t2
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 0 1 0 1 0 1
ϕ1,4 0 1 1 1 0 0 1
ϕ1,8 1 0 1 0 1 0 1
ϕ2,5 0 0 2 1 1 1 2
ϕ2,3 2 0 2 0 2 0 2
ϕ2,1 0 0 2 1 1 1 2
ϕ3,2 1 0 3 1 2 1 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 0 t8 0 t3 0 t2
ϕ1,4 0 1 t4 t 0 0 t2
ϕ1,8 t4 0 1 0 t 0 t2
ϕ2,5 0 0 t + t3 1 t2 t2 t + t3
ϕ2,3 t + t3 0 t3 + t5 0 1 + t4 0 t + t3
ϕ2,1 0 0 t5 + t7 t2 t2 1 t + t3
ϕ3,2 t2 0 t2 + t4 + t6 t t + t3 t 1 + t2 + t4

For the generic point of the hyperplane k1,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ1,8,  ϕ2,3},   {ϕ2,5,  ϕ2,1}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0111
ϕ1,42411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,8911 + 2t + 3t2 + 2t3 + t4
ϕ2,51622 + 4t + 4t2 + 4t3 + 2t4
ϕ2,3742 + 3t + 2t2
ϕ2,1822 + 4t + 2t2
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 1 0 0 0 1 1
ϕ1,4 0 1 1 1 0 0 1
ϕ1,8 0 1 1 1 0 0 1
ϕ2,5 0 2 2 2 0 0 2
ϕ2,3 0 2 0 1 1 1 2
ϕ2,1 0 2 0 1 1 1 2
ϕ3,2 0 3 1 2 1 1 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 t4 0 0 0 t t2
ϕ1,4 0 1 t4 t 0 0 t2
ϕ1,8 0 t8 1 t3 0 0 t2
ϕ2,5 0 t3 + t5 t + t3 1 + t4 0 0 t + t3
ϕ2,3 0 t5 + t7 0 t2 1 t2 t + t3
ϕ2,1 0 t + t3 0 t2 t2 1 t + t3
ϕ3,2 0 t2 + t4 + t6 t2 t + t3 t t 1 + t2 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,4,  ϕ2,3,  ϕ3,2}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,02411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,4311 + 2t
ϕ1,82411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ2,52422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3322 + t
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,21833 + 4t + 4t2 + 4t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 0 1 1 0 1 1
ϕ1,4 1 1 1 1 0 1 0
ϕ1,8 1 0 1 1 1 1 0
ϕ2,5 2 0 2 2 0 2 2
ϕ2,3 2 0 2 2 1 2 1
ϕ2,1 2 1 2 2 0 2 1
ϕ3,2 3 0 3 3 0 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 0 t8 t5 0 t t2
ϕ1,4 t8 1 t4 t 0 t3 0
ϕ1,8 t4 0 1 t3 t t5 0
ϕ2,5 t5 + t7 0 t + t3 1 + t4 0 t2 + t6 t + t3
ϕ2,3 t + t3 0 t3 + t5 t2 + t6 1 t2 + t4 t3
ϕ2,1 t3 + t5 t t5 + t7 t2 + t4 0 1 + t4 t
ϕ3,2 t2 + t4 + t6 0 t2 + t4 + t6 t + t3 + t5 0 t + t3 + t5 1 + t2 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,4,  ϕ1,8,  ϕ2,5},   {ϕ2,3,  ϕ2,1}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,02411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,4911 + 2t + 3t2 + 2t3 + t4
ϕ1,8111
ϕ2,5742 + 3t + 2t2
ϕ2,3822 + 4t + 2t2
ϕ2,11622 + 4t + 4t2 + 4t3 + 2t4
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 1 0 0 0 1 1
ϕ1,4 1 1 0 0 0 1 1
ϕ1,8 1 0 1 0 1 0 1
ϕ2,5 2 0 0 1 1 1 2
ϕ2,3 2 0 0 1 1 1 2
ϕ2,1 2 2 0 0 0 2 2
ϕ3,2 3 1 0 1 1 2 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 t4 0 0 0 t t2
ϕ1,4 t8 1 0 0 0 t3 t2
ϕ1,8 t4 0 1 0 t 0 t2
ϕ2,5 t5 + t7 0 0 1 t2 t2 t + t3
ϕ2,3 t + t3 0 0 t2 1 t2 t + t3
ϕ2,1 t3 + t5 t + t3 0 0 0 1 + t4 t + t3
ϕ3,2 t2 + t4 + t6 t2 0 t t t + t3 1 + t2 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ2,5,  ϕ3,2}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0311 + 2t
ϕ1,42411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,82411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ2,5322 + t
ϕ2,32422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,21833 + 4t + 4t2 + 4t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 1 1 0 1 1 0
ϕ1,4 0 1 1 1 1 1 0
ϕ1,8 0 1 1 0 1 1 1
ϕ2,5 0 2 2 1 2 2 1
ϕ2,3 1 2 2 0 2 2 1
ϕ2,1 0 2 2 0 2 2 2
ϕ3,2 0 3 3 0 3 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 t4 t8 0 t3 t 0
ϕ1,4 0 1 t4 t t5 t3 0
ϕ1,8 0 t8 1 0 t t5 t2
ϕ2,5 0 t3 + t5 t + t3 1 t2 + t4 t2 + t6 t3
ϕ2,3 t t5 + t7 t3 + t5 0 1 + t4 t2 + t4 t
ϕ2,1 0 t + t3 t5 + t7 0 t2 + t6 1 + t4 t + t3
ϕ3,2 0 t2 + t4 + t6 t2 + t4 + t6 0 t + t3 + t5 t + t3 + t5 1 + t2 + t4

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

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,8,  ϕ2,1,  ϕ3,2}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,02411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,42411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,8311 + 2t
ϕ2,52422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,32422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,1322 + t
ϕ3,21833 + 4t + 4t2 + 4t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 1 0 1 1 1 0
ϕ1,4 1 1 0 1 1 0 1
ϕ1,8 1 1 1 1 1 0 0
ϕ2,5 2 2 1 2 2 0 1
ϕ2,3 2 2 0 2 2 0 2
ϕ2,1 2 2 0 2 2 1 1
ϕ3,2 3 3 0 3 3 0 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ2,5) L(ϕ2,3) L(ϕ2,1) L(ϕ3,2)
ϕ1,0 1 t4 0 t5 t3 t 0
ϕ1,4 t8 1 0 t t5 0 t2
ϕ1,8 t4 t8 1 t3 t 0 0
ϕ2,5 t5 + t7 t3 + t5 t 1 + t4 t2 + t4 0 t
ϕ2,3 t + t3 t5 + t7 0 t2 + t6 1 + t4 0 t + t3
ϕ2,1 t3 + t5 t + t3 0 t2 + t4 t2 + t6 1 t3
ϕ3,2 t2 + t4 + t6 t2 + t4 + t6 0 t + t3 + t5 t + t3 + t5 0 1 + t2 + t4