The representation theory of the restricted rational Cherednik algebra for G6

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

{ϕ2,1,  ϕ2,3'},   {ϕ2,5',  ϕ2,3''},   {ϕ2,5'',  ϕ2,7}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 1 1 1 1 1 0 0 1 0 1 1
ϕ1,4 1 1 1 1 1 1 1 0 1 0 0 1 1 1
ϕ1,8 1 1 1 1 1 1 0 0 0 1 1 1 1 1
ϕ1,6 1 1 1 1 1 1 0 0 1 1 0 1 1 1
ϕ1,10 1 1 1 1 1 1 0 1 0 1 1 0 1 1
ϕ1,14 1 1 1 1 1 1 1 1 1 0 0 0 1 1
ϕ2,5'' 2 2 2 2 2 2 2 1 2 0 0 1 2 2
ϕ2,3'' 2 2 2 2 2 2 2 2 1 0 1 0 2 2
ϕ2,3' 2 2 2 2 2 2 1 0 2 1 0 2 2 2
ϕ2,7 2 2 2 2 2 2 0 1 0 2 2 1 2 2
ϕ2,1 2 2 2 2 2 2 1 2 0 1 2 0 2 2
ϕ2,5' 2 2 2 2 2 2 0 0 1 2 1 2 2 2
ϕ3,2 3 3 3 3 3 3 1 2 2 2 1 1 3 3
ϕ3,4 3 3 3 3 3 3 2 1 1 1 2 2 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 t6 t10 t14 t5 t3 0 0 t 0 t2 t4
ϕ1,4 t8 1 t4 t14 t6 t10 t 0 t3 0 0 t5 t2 t4
ϕ1,8 t4 t8 1 t10 t14 t6 0 0 0 t3 t5 t t2 t4
ϕ1,6 t6 t10 t14 1 t4 t8 0 0 t t5 0 t3 t4 t2
ϕ1,10 t14 t6 t10 t8 1 t4 0 t5 0 t t3 0 t4 t2
ϕ1,14 t10 t14 t6 t4 t8 1 t3 t t5 0 0 0 t4 t2
ϕ2,5'' t7 + t11 t3 + t11 t3 + t7 t5 + t13 t5 + t9 t + t9 1 + t4 t2 t2 + t6 0 0 t4 t + t5 2t3
ϕ2,3'' t + t9 t5 + t13 t5 + t9 t3 + t7 t7 + t11 t3 + t11 t2 + t6 1 + t4 t4 0 t2 0 2t3 t + t5
ϕ2,3' t5 + t9 t + t9 t5 + t13 t3 + t11 t3 + t7 t7 + t11 t2 0 1 + t4 t4 0 t2 + t6 2t3 t + t5
ϕ2,7 t5 + t13 t5 + t9 t + t9 t7 + t11 t3 + t11 t3 + t7 0 t4 0 1 + t4 t2 + t6 t2 2t3 t + t5
ϕ2,1 t3 + t11 t3 + t7 t7 + t11 t5 + t9 t + t9 t5 + t13 t4 t2 + t6 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 t7 + t11 t3 + t11 t + t9 t5 + t13 t5 + t9 0 0 t2 t2 + t6 t4 1 + t4 t + t5 2t3
ϕ3,2 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 t3 t + t5 t + t5 t + t5 t3 t3 1 + 2t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 t + t5 t3 t3 t3 t + t5 t + t5 2t2 + t6 1 + 2t4

Exceptional hyperplanes

k2,2
k2,1
k2,1 − k2,2
k1,1
k1,1 − k2,2
k1,1 + k2,2
k1,1 − 2k2,1 + k2,2
k1,1 − k2,1
k1,1 − k2,1 − k2,2
k1,1 − k2,1 + k2,2
k1,1 − k2,1 + 2k2,2
k1,1 + k2,1
k1,1 + k2,1 − 2k2,2
k1,1 + k2,1 − k2,2
k1,1 + k2,1 + k2,2
k1,1 + 2k2,1 − k2,2

For the generic point of the hyperplane k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

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

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,03211 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 3t8 + 2t9 + t10
ϕ1,41611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,63211 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 3t8 + 2t9 + t10
ϕ1,101611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''822 + 4t + 2t2
ϕ2,3''1622 + 4t + 4t2 + 4t3 + 2t4
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,7822 + 4t + 2t2
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'1622 + 4t + 4t2 + 4t3 + 2t4
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 0 1 1 0 1 0 1 0 0 1 0 1 1
ϕ1,4 0 1 1 0 1 1 1 0 1 0 0 0 1 1
ϕ1,8 1 0 1 1 0 1 0 0 0 0 1 1 1 1
ϕ1,6 1 0 1 1 0 1 0 0 1 0 0 1 1 1
ϕ1,10 0 1 1 0 1 1 0 0 0 1 1 0 1 1
ϕ1,14 1 0 1 1 0 1 0 1 1 0 0 0 1 1
ϕ2,5'' 1 1 2 1 1 2 1 1 2 0 0 0 2 2
ϕ2,3'' 2 0 2 2 0 2 0 2 1 0 1 0 2 2
ϕ2,3' 1 1 2 1 1 2 1 0 2 0 0 1 2 2
ϕ2,7 1 1 2 1 1 2 0 0 0 1 2 1 2 2
ϕ2,1 1 1 2 1 1 2 0 1 0 1 2 0 2 2
ϕ2,5' 2 0 2 2 0 2 0 0 1 0 1 2 2 2
ϕ3,2 2 1 3 2 1 3 0 1 2 1 1 1 3 3
ϕ3,4 2 1 3 2 1 3 1 1 1 0 2 1 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 0 t8 t6 0 t14 0 t3 0 0 t 0 t2 t4
ϕ1,4 0 1 t4 0 t6 t10 t 0 t3 0 0 0 t2 t4
ϕ1,8 t4 0 1 t10 0 t6 0 0 0 0 t5 t t2 t4
ϕ1,6 t6 0 t14 1 0 t8 0 0 t 0 0 t3 t4 t2
ϕ1,10 0 t6 t10 0 1 t4 0 0 0 t t3 0 t4 t2
ϕ1,14 t10 0 t6 t4 0 1 0 t t5 0 0 0 t4 t2
ϕ2,5'' t7 t3 t3 + t7 t5 t5 t + t9 1 t2 t2 + t6 0 0 0 t + t5 2t3
ϕ2,3'' t + t9 0 t5 + t9 t3 + t7 0 t3 + t11 0 1 + t4 t4 0 t2 0 2t3 t + t5
ϕ2,3' t5 t t5 + t13 t3 t3 t7 + t11 t2 0 1 + t4 0 0 t2 2t3 t + t5
ϕ2,7 t5 t5 t + t9 t7 t3 t3 + t7 0 0 0 1 t2 + t6 t2 2t3 t + t5
ϕ2,1 t3 t3 t7 + t11 t5 t t5 + t13 0 t2 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 0 t3 + t11 t + t9 0 t5 + t9 0 0 t2 0 t4 1 + t4 t + t5 2t3
ϕ3,2 t2 + t6 t2 t2 + t6 + t10 t4 + t8 t4 t4 + t8 + t12 0 t t + t5 t t3 t3 1 + 2t4 2t2 + t6
ϕ3,4 t4 + t8 t4 t4 + t8 + t12 t2 + t6 t2 t2 + t6 + t10 t t3 t3 0 t + t5 t 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k2,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,0,  ϕ1,8},   {ϕ2,1,  ϕ2,5'',  ϕ2,3',  ϕ2,7},   {ϕ1,6,  ϕ1,14},   {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)
ϕ1,01611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,83211 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 3t8 + 2t9 + t10
ϕ1,61611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,143211 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 3t8 + 2t9 + t10
ϕ2,5''1622 + 4t + 4t2 + 4t3 + 2t4
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'822 + 4t + 2t2
ϕ2,71622 + 4t + 4t2 + 4t3 + 2t4
ϕ2,1822 + 4t + 2t2
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 0 1 1 0 0 1 0 0 1 0 1 1
ϕ1,4 0 1 1 0 1 1 1 0 0 0 0 1 1 1
ϕ1,8 0 1 1 0 1 1 0 0 0 1 0 1 1 1
ϕ1,6 1 1 0 1 1 0 0 0 1 0 0 1 1 1
ϕ1,10 0 1 1 0 1 1 0 1 0 1 0 0 1 1
ϕ1,14 0 1 1 0 1 1 1 1 0 0 0 0 1 1
ϕ2,5'' 0 2 2 0 2 2 2 1 0 0 0 1 2 2
ϕ2,3'' 1 2 1 1 2 1 1 2 0 0 1 0 2 2
ϕ2,3' 1 2 1 1 2 1 1 0 1 0 0 2 2 2
ϕ2,7 0 2 2 0 2 2 0 1 0 2 0 1 2 2
ϕ2,1 1 2 1 1 2 1 0 2 0 1 1 0 2 2
ϕ2,5' 1 2 1 1 2 1 0 0 1 1 0 2 2 2
ϕ3,2 1 3 2 1 3 2 1 2 1 1 0 1 3 3
ϕ3,4 1 3 2 1 3 2 1 1 0 1 1 2 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 0 t6 t10 0 0 t3 0 0 t 0 t2 t4
ϕ1,4 0 1 t4 0 t6 t10 t 0 0 0 0 t5 t2 t4
ϕ1,8 0 t8 1 0 t14 t6 0 0 0 t3 0 t t2 t4
ϕ1,6 t6 t10 0 1 t4 0 0 0 t 0 0 t3 t4 t2
ϕ1,10 0 t6 t10 0 1 t4 0 t5 0 t 0 0 t4 t2
ϕ1,14 0 t14 t6 0 t8 1 t3 t 0 0 0 0 t4 t2
ϕ2,5'' 0 t3 + t11 t3 + t7 0 t5 + t9 t + t9 1 + t4 t2 0 0 0 t4 t + t5 2t3
ϕ2,3'' t t5 + t13 t5 t3 t7 + t11 t3 t2 1 + t4 0 0 t2 0 2t3 t + t5
ϕ2,3' t5 t + t9 t5 t3 t3 + t7 t7 t2 0 1 0 0 t2 + t6 2t3 t + t5
ϕ2,7 0 t5 + t9 t + t9 0 t3 + t11 t3 + t7 0 t4 0 1 + t4 0 t2 2t3 t + t5
ϕ2,1 t3 t3 + t7 t7 t5 t + t9 t5 0 t2 + t6 0 t2 1 0 t + t5 2t3
ϕ2,5' t3 t7 + t11 t3 t t5 + t13 t5 0 0 t2 t2 0 1 + t4 t + t5 2t3
ϕ3,2 t2 t2 + t6 + t10 t2 + t6 t4 t4 + t8 + t12 t4 + t8 t3 t + t5 t t 0 t3 1 + 2t4 2t2 + t6
ϕ3,4 t4 t4 + t8 + t12 t4 + t8 t2 t2 + t6 + t10 t2 + t6 t t3 0 t3 t t + t5 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,4,  ϕ1,8},   {ϕ2,5'',  ϕ2,7},   {ϕ2,1,  ϕ2,5',  ϕ2,3'',  ϕ2,3'},   {ϕ1,10,  ϕ1,14}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,43211 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 3t8 + 2t9 + t10
ϕ1,81611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,103211 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 3t8 + 2t9 + t10
ϕ1,141611 + 2t + 3t2 + 4t3 + 3t4 + 2t5 + t6
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''822 + 4t + 2t2
ϕ2,3'1622 + 4t + 4t2 + 4t3 + 2t4
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,11622 + 4t + 4t2 + 4t3 + 2t4
ϕ2,5'822 + 4t + 2t2
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 0 1 1 0 1 0 0 0 1 0 1 1
ϕ1,4 1 1 0 1 1 0 1 0 1 0 0 0 1 1
ϕ1,8 1 0 1 1 0 1 0 0 0 1 0 1 1 1
ϕ1,6 1 1 0 1 1 0 0 0 1 1 0 0 1 1
ϕ1,10 1 1 0 1 1 0 0 0 0 1 1 0 1 1
ϕ1,14 1 0 1 1 0 1 1 1 0 0 0 0 1 1
ϕ2,5'' 2 1 1 2 1 1 2 1 1 0 0 0 2 2
ϕ2,3'' 2 1 1 2 1 1 2 1 0 0 1 0 2 2
ϕ2,3' 2 2 0 2 2 0 1 0 2 1 0 0 2 2
ϕ2,7 2 1 1 2 1 1 0 0 0 2 1 1 2 2
ϕ2,1 2 2 0 2 2 0 1 0 0 1 2 0 2 2
ϕ2,5' 2 1 1 2 1 1 0 0 1 2 0 1 2 2
ϕ3,2 3 2 1 3 2 1 1 1 1 2 1 0 3 3
ϕ3,4 3 2 1 3 2 1 2 0 1 1 1 1 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 0 t6 t10 0 t5 0 0 0 t 0 t2 t4
ϕ1,4 t8 1 0 t14 t6 0 t 0 t3 0 0 0 t2 t4
ϕ1,8 t4 0 1 t10 0 t6 0 0 0 t3 0 t t2 t4
ϕ1,6 t6 t10 0 1 t4 0 0 0 t t5 0 0 t4 t2
ϕ1,10 t14 t6 0 t8 1 0 0 0 0 t t3 0 t4 t2
ϕ1,14 t10 0 t6 t4 0 1 t3 t 0 0 0 0 t4 t2
ϕ2,5'' t7 + t11 t3 t3 t5 + t13 t5 t 1 + t4 t2 t2 0 0 0 t + t5 2t3
ϕ2,3'' t + t9 t5 t5 t3 + t7 t7 t3 t2 + t6 1 0 0 t2 0 2t3 t + t5
ϕ2,3' t5 + t9 t + t9 0 t3 + t11 t3 + t7 0 t2 0 1 + t4 t4 0 0 2t3 t + t5
ϕ2,7 t5 + t13 t5 t t7 + t11 t3 t3 0 0 0 1 + t4 t2 t2 2t3 t + t5
ϕ2,1 t3 + t11 t3 + t7 0 t5 + t9 t + t9 0 t4 0 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 t7 t3 t + t9 t5 t5 0 0 t2 t2 + t6 0 1 t + t5 2t3
ϕ3,2 t2 + t6 + t10 t2 + t6 t2 t4 + t8 + t12 t4 + t8 t4 t3 t t t + t5 t3 0 1 + 2t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 t4 + t8 t4 t2 + t6 + t10 t2 + t6 t2 t + t5 0 t3 t3 t t 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k1,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,4,  ϕ1,10},   {ϕ1,8,  ϕ1,14},   {ϕ1,0,  ϕ1,6},   {ϕ2,5',  ϕ2,3''},   {ϕ2,5'',  ϕ2,7},   {ϕ2,1,  ϕ2,3'},   {ϕ3,2,  ϕ3,4}

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
ϕ1,62411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,102411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ1,142411 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 3t6 + 2t7 + t8
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,22433 + 6t + 6t2 + 6t3 + 3t4
ϕ3,42433 + 6t + 6t2 + 6t3 + 3t4

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 0 0 0 1 1 0 0 1 0 1 0
ϕ1,4 1 1 1 0 0 0 1 0 1 0 0 1 1 0
ϕ1,8 1 1 1 0 0 0 0 0 0 1 1 1 1 0
ϕ1,6 0 0 0 1 1 1 0 0 1 1 0 1 0 1
ϕ1,10 0 0 0 1 1 1 0 1 0 1 1 0 0 1
ϕ1,14 0 0 0 1 1 1 1 1 1 0 0 0 0 1
ϕ2,5'' 1 1 1 1 1 1 2 1 2 0 0 1 1 1
ϕ2,3'' 1 1 1 1 1 1 2 2 1 0 1 0 1 1
ϕ2,3' 1 1 1 1 1 1 1 0 2 1 0 2 1 1
ϕ2,7 1 1 1 1 1 1 0 1 0 2 2 1 1 1
ϕ2,1 1 1 1 1 1 1 1 2 0 1 2 0 1 1
ϕ2,5' 1 1 1 1 1 1 0 0 1 2 1 2 1 1
ϕ3,2 2 2 2 1 1 1 1 2 2 2 1 1 2 1
ϕ3,4 1 1 1 2 2 2 2 1 1 1 2 2 1 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 0 0 0 t5 t3 0 0 t 0 t2 0
ϕ1,4 t8 1 t4 0 0 0 t 0 t3 0 0 t5 t2 0
ϕ1,8 t4 t8 1 0 0 0 0 0 0 t3 t5 t t2 0
ϕ1,6 0 0 0 1 t4 t8 0 0 t t5 0 t3 0 t2
ϕ1,10 0 0 0 t8 1 t4 0 t5 0 t t3 0 0 t2
ϕ1,14 0 0 0 t4 t8 1 t3 t t5 0 0 0 0 t2
ϕ2,5'' t7 t3 t3 t5 t5 t 1 + t4 t2 t2 + t6 0 0 t4 t t3
ϕ2,3'' t t5 t5 t3 t7 t3 t2 + t6 1 + t4 t4 0 t2 0 t3 t
ϕ2,3' t5 t t5 t3 t3 t7 t2 0 1 + t4 t4 0 t2 + t6 t3 t
ϕ2,7 t5 t5 t t7 t3 t3 0 t4 0 1 + t4 t2 + t6 t2 t3 t
ϕ2,1 t3 t3 t7 t5 t t5 t4 t2 + t6 0 t2 1 + t4 0 t t3
ϕ2,5' t3 t7 t3 t t5 t5 0 0 t2 t2 + t6 t4 1 + t4 t t3
ϕ3,2 t2 + t6 t2 + t6 t2 + t6 t4 t4 t4 t3 t + t5 t + t5 t + t5 t3 t3 1 + t4 t2
ϕ3,4 t4 t4 t4 t2 + t6 t2 + t6 t2 + t6 t + t5 t3 t3 t3 t + t5 t + t5 t2 1 + t4

For the generic point of the hyperplane k1,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,5',  ϕ2,3''},   {ϕ2,1,  ϕ1,4,  ϕ1,6,  ϕ2,3'},   {ϕ2,5'',  ϕ2,7}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,4111
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,6911 + 2t + 3t2 + 2t3 + t4
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'742 + 3t + 2t2
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 0 1 0 1 1 1 1 0 0 1 0 1 1
ϕ1,4 1 1 1 0 1 1 1 0 0 0 0 1 1 1
ϕ1,8 1 0 1 0 1 1 0 0 0 1 1 1 1 1
ϕ1,6 1 0 1 1 1 1 0 0 0 1 0 1 1 1
ϕ1,10 1 0 1 0 1 1 0 1 0 1 1 0 1 1
ϕ1,14 1 0 1 1 1 1 1 1 0 0 0 0 1 1
ϕ2,5'' 2 0 2 0 2 2 2 1 1 0 0 1 2 2
ϕ2,3'' 2 0 2 1 2 2 2 2 0 0 1 0 2 2
ϕ2,3' 2 0 2 0 2 2 1 0 1 1 0 2 2 2
ϕ2,7 2 0 2 0 2 2 0 1 0 2 2 1 2 2
ϕ2,1 2 0 2 0 2 2 1 2 0 1 2 0 2 2
ϕ2,5' 2 0 2 1 2 2 0 0 0 2 1 2 2 2
ϕ3,2 3 0 3 0 3 3 1 2 1 2 1 1 3 3
ϕ3,4 3 0 3 1 3 3 2 1 0 1 2 2 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 0 t8 0 t10 t14 t5 t3 0 0 t 0 t2 t4
ϕ1,4 t8 1 t4 0 t6 t10 t 0 0 0 0 t5 t2 t4
ϕ1,8 t4 0 1 0 t14 t6 0 0 0 t3 t5 t t2 t4
ϕ1,6 t6 0 t14 1 t4 t8 0 0 0 t5 0 t3 t4 t2
ϕ1,10 t14 0 t10 0 1 t4 0 t5 0 t t3 0 t4 t2
ϕ1,14 t10 0 t6 t4 t8 1 t3 t 0 0 0 0 t4 t2
ϕ2,5'' t7 + t11 0 t3 + t7 0 t5 + t9 t + t9 1 + t4 t2 t2 0 0 t4 t + t5 2t3
ϕ2,3'' t + t9 0 t5 + t9 t3 t7 + t11 t3 + t11 t2 + t6 1 + t4 0 0 t2 0 2t3 t + t5
ϕ2,3' t5 + t9 0 t5 + t13 0 t3 + t7 t7 + t11 t2 0 1 t4 0 t2 + t6 2t3 t + t5
ϕ2,7 t5 + t13 0 t + t9 0 t3 + t11 t3 + t7 0 t4 0 1 + t4 t2 + t6 t2 2t3 t + t5
ϕ2,1 t3 + t11 0 t7 + t11 0 t + t9 t5 + t13 t4 t2 + t6 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 0 t3 + t11 t t5 + t13 t5 + t9 0 0 0 t2 + t6 t4 1 + t4 t + t5 2t3
ϕ3,2 t2 + t6 + t10 0 t2 + t6 + t10 0 t4 + t8 + t12 t4 + t8 + t12 t3 t + t5 t t + t5 t3 t3 1 + 2t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 0 t4 + t8 + t12 t2 t2 + t6 + t10 t2 + t6 + t10 t + t5 t3 0 t3 t + t5 t + t5 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k1,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,5',  ϕ2,3''},   {ϕ2,1,  ϕ1,0,  ϕ1,10,  ϕ2,3'},   {ϕ2,5'',  ϕ2,7}

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
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,10111
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,1742 + 3t + 2t2
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 1 0 1 1 1 0 0 0 0 1 1
ϕ1,4 0 1 1 1 0 1 1 0 1 0 0 1 1 1
ϕ1,8 1 1 1 1 0 1 0 0 0 1 0 1 1 1
ϕ1,6 0 1 1 1 0 1 0 0 1 1 0 1 1 1
ϕ1,10 0 1 1 1 1 1 0 1 0 1 0 0 1 1
ϕ1,14 0 1 1 1 0 1 1 1 1 0 0 0 1 1
ϕ2,5'' 0 2 2 2 0 2 2 1 2 0 0 1 2 2
ϕ2,3'' 1 2 2 2 0 2 2 2 1 0 0 0 2 2
ϕ2,3' 0 2 2 2 0 2 1 0 2 1 0 2 2 2
ϕ2,7 0 2 2 2 0 2 0 1 0 2 1 1 2 2
ϕ2,1 0 2 2 2 0 2 1 2 0 1 1 0 2 2
ϕ2,5' 1 2 2 2 0 2 0 0 1 2 0 2 2 2
ϕ3,2 1 3 3 3 0 3 1 2 2 2 0 1 3 3
ϕ3,4 0 3 3 3 0 3 2 1 1 1 1 2 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 t6 0 t14 t5 t3 0 0 0 0 t2 t4
ϕ1,4 0 1 t4 t14 0 t10 t 0 t3 0 0 t5 t2 t4
ϕ1,8 t4 t8 1 t10 0 t6 0 0 0 t3 0 t t2 t4
ϕ1,6 0 t10 t14 1 0 t8 0 0 t t5 0 t3 t4 t2
ϕ1,10 0 t6 t10 t8 1 t4 0 t5 0 t 0 0 t4 t2
ϕ1,14 0 t14 t6 t4 0 1 t3 t t5 0 0 0 t4 t2
ϕ2,5'' 0 t3 + t11 t3 + t7 t5 + t13 0 t + t9 1 + t4 t2 t2 + t6 0 0 t4 t + t5 2t3
ϕ2,3'' t t5 + t13 t5 + t9 t3 + t7 0 t3 + t11 t2 + t6 1 + t4 t4 0 0 0 2t3 t + t5
ϕ2,3' 0 t + t9 t5 + t13 t3 + t11 0 t7 + t11 t2 0 1 + t4 t4 0 t2 + t6 2t3 t + t5
ϕ2,7 0 t5 + t9 t + t9 t7 + t11 0 t3 + t7 0 t4 0 1 + t4 t2 t2 2t3 t + t5
ϕ2,1 0 t3 + t7 t7 + t11 t5 + t9 0 t5 + t13 t4 t2 + t6 0 t2 1 0 t + t5 2t3
ϕ2,5' t3 t7 + t11 t3 + t11 t + t9 0 t5 + t9 0 0 t2 t2 + t6 0 1 + t4 t + t5 2t3
ϕ3,2 t2 t2 + t6 + t10 t2 + t6 + t10 t4 + t8 + t12 0 t4 + t8 + t12 t3 t + t5 t + t5 t + t5 0 t3 1 + 2t4 2t2 + t6
ϕ3,4 0 t4 + t8 + t12 t4 + t8 + t12 t2 + t6 + t10 0 t2 + t6 + t10 t + t5 t3 t3 t3 t t + t5 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k1,1 − 2k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ3,2,  ϕ1,8,  ϕ2,3'},   {ϕ2,5',  ϕ2,3''},   {ϕ2,5'',  ϕ2,7}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,8311 + 2t
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'322 + t
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12122 + 4t + 4t2 + 4t3 + 4t4 + 3t5
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,22133 + 4t + 4t2 + 4t3 + 4t4 + 2t5
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 0 1 1 1 1 1 0 0 1 0 0 1
ϕ1,4 1 1 0 1 1 1 1 0 0 0 0 1 1 1
ϕ1,8 1 1 1 1 1 1 0 0 0 1 0 1 0 1
ϕ1,6 1 1 0 1 1 1 0 0 1 1 0 1 0 1
ϕ1,10 1 1 0 1 1 1 0 1 0 1 1 0 0 1
ϕ1,14 1 1 0 1 1 1 1 1 0 0 0 0 1 1
ϕ2,5'' 2 2 0 2 2 2 2 1 0 0 0 1 2 2
ϕ2,3'' 2 2 0 2 2 2 2 2 0 0 1 0 1 2
ϕ2,3' 2 2 0 2 2 2 1 0 1 1 0 2 1 2
ϕ2,7 2 2 1 2 2 2 0 1 0 2 1 1 0 2
ϕ2,1 2 2 0 2 2 2 1 2 0 1 2 0 0 2
ϕ2,5' 2 2 0 2 2 2 0 0 0 2 1 2 1 2
ϕ3,2 3 3 0 3 3 3 1 2 0 2 1 1 2 3
ϕ3,4 3 3 0 3 3 3 2 1 0 1 2 2 1 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 0 t6 t10 t14 t5 t3 0 0 t 0 0 t4
ϕ1,4 t8 1 0 t14 t6 t10 t 0 0 0 0 t5 t2 t4
ϕ1,8 t4 t8 1 t10 t14 t6 0 0 0 t3 0 t 0 t4
ϕ1,6 t6 t10 0 1 t4 t8 0 0 t t5 0 t3 0 t2
ϕ1,10 t14 t6 0 t8 1 t4 0 t5 0 t t3 0 0 t2
ϕ1,14 t10 t14 0 t4 t8 1 t3 t 0 0 0 0 t4 t2
ϕ2,5'' t7 + t11 t3 + t11 0 t5 + t13 t5 + t9 t + t9 1 + t4 t2 0 0 0 t4 t + t5 2t3
ϕ2,3'' t + t9 t5 + t13 0 t3 + t7 t7 + t11 t3 + t11 t2 + t6 1 + t4 0 0 t2 0 t3 t + t5
ϕ2,3' t5 + t9 t + t9 0 t3 + t11 t3 + t7 t7 + t11 t2 0 1 t4 0 t2 + t6 t3 t + t5
ϕ2,7 t5 + t13 t5 + t9 t t7 + t11 t3 + t11 t3 + t7 0 t4 0 1 + t4 t2 t2 0 t + t5
ϕ2,1 t3 + t11 t3 + t7 0 t5 + t9 t + t9 t5 + t13 t4 t2 + t6 0 t2 1 + t4 0 0 2t3
ϕ2,5' t3 + t7 t7 + t11 0 t + t9 t5 + t13 t5 + t9 0 0 0 t2 + t6 t4 1 + t4 t 2t3
ϕ3,2 t2 + t6 + t10 t2 + t6 + t10 0 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 t3 t + t5 0 t + t5 t3 t3 1 + t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 t4 + t8 + t12 0 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 t + t5 t3 0 t3 t + t5 t + t5 t2 1 + 2t4

For the generic point of the hyperplane k1,1 − k2,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ2,3'},   {ϕ2,5',  ϕ1,8,  ϕ1,6,  ϕ2,3''},   {ϕ2,5'',  ϕ2,7}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,8911 + 2t + 3t2 + 2t3 + t4
ϕ1,6111
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'742 + 3t + 2t2
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 0 0 1 1 1 1 0 0 1 0 1 1
ϕ1,4 1 1 1 0 1 1 1 0 1 0 0 0 1 1
ϕ1,8 1 1 1 0 1 1 0 0 0 1 1 0 1 1
ϕ1,6 1 1 0 1 1 1 0 0 1 1 0 0 1 1
ϕ1,10 1 1 0 0 1 1 0 1 0 1 1 0 1 1
ϕ1,14 1 1 0 0 1 1 1 1 1 0 0 0 1 1
ϕ2,5'' 2 2 1 0 2 2 2 1 2 0 0 0 2 2
ϕ2,3'' 2 2 0 0 2 2 2 2 1 0 1 0 2 2
ϕ2,3' 2 2 0 0 2 2 1 0 2 1 0 1 2 2
ϕ2,7 2 2 1 0 2 2 0 1 0 2 2 0 2 2
ϕ2,1 2 2 0 0 2 2 1 2 0 1 2 0 2 2
ϕ2,5' 2 2 0 0 2 2 0 0 1 2 1 1 2 2
ϕ3,2 3 3 1 0 3 3 1 2 2 2 1 0 3 3
ϕ3,4 3 3 0 0 3 3 2 1 1 1 2 1 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 0 0 t10 t14 t5 t3 0 0 t 0 t2 t4
ϕ1,4 t8 1 t4 0 t6 t10 t 0 t3 0 0 0 t2 t4
ϕ1,8 t4 t8 1 0 t14 t6 0 0 0 t3 t5 0 t2 t4
ϕ1,6 t6 t10 0 1 t4 t8 0 0 t t5 0 0 t4 t2
ϕ1,10 t14 t6 0 0 1 t4 0 t5 0 t t3 0 t4 t2
ϕ1,14 t10 t14 0 0 t8 1 t3 t t5 0 0 0 t4 t2
ϕ2,5'' t7 + t11 t3 + t11 t3 0 t5 + t9 t + t9 1 + t4 t2 t2 + t6 0 0 0 t + t5 2t3
ϕ2,3'' t + t9 t5 + t13 0 0 t7 + t11 t3 + t11 t2 + t6 1 + t4 t4 0 t2 0 2t3 t + t5
ϕ2,3' t5 + t9 t + t9 0 0 t3 + t7 t7 + t11 t2 0 1 + t4 t4 0 t2 2t3 t + t5
ϕ2,7 t5 + t13 t5 + t9 t 0 t3 + t11 t3 + t7 0 t4 0 1 + t4 t2 + t6 0 2t3 t + t5
ϕ2,1 t3 + t11 t3 + t7 0 0 t + t9 t5 + t13 t4 t2 + t6 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 t7 + t11 0 0 t5 + t13 t5 + t9 0 0 t2 t2 + t6 t4 1 t + t5 2t3
ϕ3,2 t2 + t6 + t10 t2 + t6 + t10 t2 0 t4 + t8 + t12 t4 + t8 + t12 t3 t + t5 t + t5 t + t5 t3 0 1 + 2t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 t4 + t8 + t12 0 0 t2 + t6 + t10 t2 + t6 + t10 t + t5 t3 t3 t3 t + t5 t 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k1,1 − k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ3,4,  ϕ1,6,  ϕ2,5'',  ϕ2,7},   {ϕ2,1,  ϕ2,3'},   {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)
ϕ1,04811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,6311 + 2t
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''322 + t
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72122 + 4t + 4t2 + 4t3 + 4t4 + 3t5
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,42133 + 4t + 4t2 + 4t3 + 4t4 + 2t5

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 0 1 1 0 1 0 0 1 0 1 1
ϕ1,4 1 1 1 0 1 1 1 0 1 0 0 1 1 0
ϕ1,8 1 1 1 0 1 1 0 0 0 1 1 1 1 0
ϕ1,6 1 1 1 1 1 1 0 0 1 0 0 1 1 0
ϕ1,10 1 1 1 0 1 1 0 1 0 1 1 0 1 0
ϕ1,14 1 1 1 0 1 1 0 1 1 0 0 0 1 1
ϕ2,5'' 2 2 2 0 2 2 1 1 2 0 0 1 2 1
ϕ2,3'' 2 2 2 0 2 2 0 2 1 0 1 0 2 2
ϕ2,3' 2 2 2 0 2 2 0 0 2 1 0 2 2 1
ϕ2,7 2 2 2 0 2 2 0 1 0 2 2 1 2 0
ϕ2,1 2 2 2 0 2 2 0 2 0 1 2 0 2 1
ϕ2,5' 2 2 2 1 2 2 0 0 1 1 1 2 2 0
ϕ3,2 3 3 3 0 3 3 0 2 2 2 1 1 3 1
ϕ3,4 3 3 3 0 3 3 0 1 1 1 2 2 3 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 0 t10 t14 0 t3 0 0 t 0 t2 t4
ϕ1,4 t8 1 t4 0 t6 t10 t 0 t3 0 0 t5 t2 0
ϕ1,8 t4 t8 1 0 t14 t6 0 0 0 t3 t5 t t2 0
ϕ1,6 t6 t10 t14 1 t4 t8 0 0 t 0 0 t3 t4 0
ϕ1,10 t14 t6 t10 0 1 t4 0 t5 0 t t3 0 t4 0
ϕ1,14 t10 t14 t6 0 t8 1 0 t t5 0 0 0 t4 t2
ϕ2,5'' t7 + t11 t3 + t11 t3 + t7 0 t5 + t9 t + t9 1 t2 t2 + t6 0 0 t4 t + t5 t3
ϕ2,3'' t + t9 t5 + t13 t5 + t9 0 t7 + t11 t3 + t11 0 1 + t4 t4 0 t2 0 2t3 t + t5
ϕ2,3' t5 + t9 t + t9 t5 + t13 0 t3 + t7 t7 + t11 0 0 1 + t4 t4 0 t2 + t6 2t3 t
ϕ2,7 t5 + t13 t5 + t9 t + t9 0 t3 + t11 t3 + t7 0 t4 0 1 + t4 t2 + t6 t2 2t3 0
ϕ2,1 t3 + t11 t3 + t7 t7 + t11 0 t + t9 t5 + t13 0 t2 + t6 0 t2 1 + t4 0 t + t5 t3
ϕ2,5' t3 + t7 t7 + t11 t3 + t11 t t5 + t13 t5 + t9 0 0 t2 t2 t4 1 + t4 t + t5 0
ϕ3,2 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 0 t4 + t8 + t12 t4 + t8 + t12 0 t + t5 t + t5 t + t5 t3 t3 1 + 2t4 t2
ϕ3,4 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 0 t2 + t6 + t10 t2 + t6 + t10 0 t3 t3 t3 t + t5 t + t5 2t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 − k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ2,3'},   {ϕ2,5',  ϕ2,3''},   {ϕ1,8,  ϕ1,10,  ϕ2,5'',  ϕ2,7}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,8111
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,10911 + 2t + 3t2 + 2t3 + t4
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,7742 + 3t + 2t2
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 0 1 0 1 1 1 0 0 1 0 1 1
ϕ1,4 1 1 0 1 0 1 1 0 1 0 0 1 1 1
ϕ1,8 1 1 1 1 0 1 0 0 0 0 1 1 1 1
ϕ1,6 1 1 0 1 1 1 0 0 1 0 0 1 1 1
ϕ1,10 1 1 0 1 1 1 0 1 0 0 1 0 1 1
ϕ1,14 1 1 0 1 0 1 1 1 1 0 0 0 1 1
ϕ2,5'' 2 2 0 2 0 2 2 1 2 0 0 1 2 2
ϕ2,3'' 2 2 0 2 0 2 2 2 1 0 1 0 2 2
ϕ2,3' 2 2 0 2 1 2 1 0 2 0 0 2 2 2
ϕ2,7 2 2 0 2 0 2 0 1 0 1 2 1 2 2
ϕ2,1 2 2 0 2 1 2 1 2 0 0 2 0 2 2
ϕ2,5' 2 2 0 2 0 2 0 0 1 1 1 2 2 2
ϕ3,2 3 3 0 3 0 3 1 2 2 1 1 1 3 3
ϕ3,4 3 3 0 3 1 3 2 1 1 0 2 2 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 0 t6 0 t14 t5 t3 0 0 t 0 t2 t4
ϕ1,4 t8 1 0 t14 0 t10 t 0 t3 0 0 t5 t2 t4
ϕ1,8 t4 t8 1 t10 0 t6 0 0 0 0 t5 t t2 t4
ϕ1,6 t6 t10 0 1 t4 t8 0 0 t 0 0 t3 t4 t2
ϕ1,10 t14 t6 0 t8 1 t4 0 t5 0 0 t3 0 t4 t2
ϕ1,14 t10 t14 0 t4 0 1 t3 t t5 0 0 0 t4 t2
ϕ2,5'' t7 + t11 t3 + t11 0 t5 + t13 0 t + t9 1 + t4 t2 t2 + t6 0 0 t4 t + t5 2t3
ϕ2,3'' t + t9 t5 + t13 0 t3 + t7 0 t3 + t11 t2 + t6 1 + t4 t4 0 t2 0 2t3 t + t5
ϕ2,3' t5 + t9 t + t9 0 t3 + t11 t3 t7 + t11 t2 0 1 + t4 0 0 t2 + t6 2t3 t + t5
ϕ2,7 t5 + t13 t5 + t9 0 t7 + t11 0 t3 + t7 0 t4 0 1 t2 + t6 t2 2t3 t + t5
ϕ2,1 t3 + t11 t3 + t7 0 t5 + t9 t t5 + t13 t4 t2 + t6 0 0 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 t7 + t11 0 t + t9 0 t5 + t9 0 0 t2 t2 t4 1 + t4 t + t5 2t3
ϕ3,2 t2 + t6 + t10 t2 + t6 + t10 0 t4 + t8 + t12 0 t4 + t8 + t12 t3 t + t5 t + t5 t t3 t3 1 + 2t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 t4 + t8 + t12 0 t2 + t6 + t10 t2 t2 + t6 + t10 t + t5 t3 t3 0 t + t5 t + t5 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k1,1 − k2,1 + 2k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,5',  ϕ3,4,  ϕ1,10,  ϕ2,3''},   {ϕ2,5'',  ϕ2,7},   {ϕ2,1,  ϕ2,3'}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,10311 + 2t
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2122 + 4t + 4t2 + 4t3 + 4t4 + 3t5
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'322 + t
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,42133 + 4t + 4t2 + 4t3 + 4t4 + 2t5

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 1 0 1 1 1 0 0 1 0 1 0
ϕ1,4 1 1 1 1 0 1 1 0 1 0 0 0 1 1
ϕ1,8 1 1 1 1 0 1 0 0 0 1 1 1 1 0
ϕ1,6 1 1 1 1 0 1 0 0 1 1 0 0 1 1
ϕ1,10 1 1 1 1 1 1 0 0 0 1 1 0 1 0
ϕ1,14 1 1 1 1 0 1 1 1 1 0 0 0 1 0
ϕ2,5'' 2 2 2 2 0 2 2 1 2 0 0 0 2 1
ϕ2,3'' 2 2 2 2 0 2 2 2 1 0 1 0 2 0
ϕ2,3' 2 2 2 2 0 2 1 0 2 1 0 0 2 2
ϕ2,7 2 2 2 2 0 2 0 1 0 2 2 0 2 1
ϕ2,1 2 2 2 2 1 2 1 1 0 1 2 0 2 0
ϕ2,5' 2 2 2 2 0 2 0 0 1 2 1 1 2 1
ϕ3,2 3 3 3 3 0 3 1 2 2 2 1 0 3 1
ϕ3,4 3 3 3 3 0 3 2 1 1 1 2 0 3 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 t6 0 t14 t5 t3 0 0 t 0 t2 0
ϕ1,4 t8 1 t4 t14 0 t10 t 0 t3 0 0 0 t2 t4
ϕ1,8 t4 t8 1 t10 0 t6 0 0 0 t3 t5 t t2 0
ϕ1,6 t6 t10 t14 1 0 t8 0 0 t t5 0 0 t4 t2
ϕ1,10 t14 t6 t10 t8 1 t4 0 0 0 t t3 0 t4 0
ϕ1,14 t10 t14 t6 t4 0 1 t3 t t5 0 0 0 t4 0
ϕ2,5'' t7 + t11 t3 + t11 t3 + t7 t5 + t13 0 t + t9 1 + t4 t2 t2 + t6 0 0 0 t + t5 t3
ϕ2,3'' t + t9 t5 + t13 t5 + t9 t3 + t7 0 t3 + t11 t2 + t6 1 + t4 t4 0 t2 0 2t3 0
ϕ2,3' t5 + t9 t + t9 t5 + t13 t3 + t11 0 t7 + t11 t2 0 1 + t4 t4 0 0 2t3 t + t5
ϕ2,7 t5 + t13 t5 + t9 t + t9 t7 + t11 0 t3 + t7 0 t4 0 1 + t4 t2 + t6 0 2t3 t
ϕ2,1 t3 + t11 t3 + t7 t7 + t11 t5 + t9 t t5 + t13 t4 t2 0 t2 1 + t4 0 t + t5 0
ϕ2,5' t3 + t7 t7 + t11 t3 + t11 t + t9 0 t5 + t9 0 0 t2 t2 + t6 t4 1 t + t5 t3
ϕ3,2 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 t4 + t8 + t12 0 t4 + t8 + t12 t3 t + t5 t + t5 t + t5 t3 0 1 + 2t4 t2
ϕ3,4 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 t2 + t6 + t10 0 t2 + t6 + t10 t + t5 t3 t3 t3 t + t5 0 2t2 + t6 1 + t4

For the generic point of the hyperplane k1,1 + k2,1

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ2,3'},   {ϕ1,0,  ϕ2,5',  ϕ1,14,  ϕ2,3''},   {ϕ2,5'',  ϕ2,7}

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

ϕ dim L(ϕ) [Δ(ϕ) : L(ϕ)] PL(ϕ)
ϕ1,0111
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,14911 + 2t + 3t2 + 2t3 + t4
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''742 + 3t + 2t2
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 1 1 0 1 0 0 0 1 0 1 1
ϕ1,4 0 1 1 1 1 0 1 0 1 0 0 1 1 1
ϕ1,8 0 1 1 1 1 0 0 0 0 1 1 1 1 1
ϕ1,6 0 1 1 1 1 0 0 0 1 1 0 1 1 1
ϕ1,10 0 1 1 1 1 1 0 0 0 1 1 0 1 1
ϕ1,14 0 1 1 1 1 1 1 0 1 0 0 0 1 1
ϕ2,5'' 0 2 2 2 2 1 2 0 2 0 0 1 2 2
ϕ2,3'' 0 2 2 2 2 0 2 1 1 0 1 0 2 2
ϕ2,3' 0 2 2 2 2 0 1 0 2 1 0 2 2 2
ϕ2,7 0 2 2 2 2 1 0 0 0 2 2 1 2 2
ϕ2,1 0 2 2 2 2 0 1 1 0 1 2 0 2 2
ϕ2,5' 0 2 2 2 2 0 0 0 1 2 1 2 2 2
ϕ3,2 0 3 3 3 3 0 1 1 2 2 1 1 3 3
ϕ3,4 0 3 3 3 3 1 2 0 1 1 2 2 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 t6 t10 0 t5 0 0 0 t 0 t2 t4
ϕ1,4 0 1 t4 t14 t6 0 t 0 t3 0 0 t5 t2 t4
ϕ1,8 0 t8 1 t10 t14 0 0 0 0 t3 t5 t t2 t4
ϕ1,6 0 t10 t14 1 t4 0 0 0 t t5 0 t3 t4 t2
ϕ1,10 0 t6 t10 t8 1 t4 0 0 0 t t3 0 t4 t2
ϕ1,14 0 t14 t6 t4 t8 1 t3 0 t5 0 0 0 t4 t2
ϕ2,5'' 0 t3 + t11 t3 + t7 t5 + t13 t5 + t9 t 1 + t4 0 t2 + t6 0 0 t4 t + t5 2t3
ϕ2,3'' 0 t5 + t13 t5 + t9 t3 + t7 t7 + t11 0 t2 + t6 1 t4 0 t2 0 2t3 t + t5
ϕ2,3' 0 t + t9 t5 + t13 t3 + t11 t3 + t7 0 t2 0 1 + t4 t4 0 t2 + t6 2t3 t + t5
ϕ2,7 0 t5 + t9 t + t9 t7 + t11 t3 + t11 t3 0 0 0 1 + t4 t2 + t6 t2 2t3 t + t5
ϕ2,1 0 t3 + t7 t7 + t11 t5 + t9 t + t9 0 t4 t2 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' 0 t7 + t11 t3 + t11 t + t9 t5 + t13 0 0 0 t2 t2 + t6 t4 1 + t4 t + t5 2t3
ϕ3,2 0 t2 + t6 + t10 t2 + t6 + t10 t4 + t8 + t12 t4 + t8 + t12 0 t3 t t + t5 t + t5 t3 t3 1 + 2t4 2t2 + t6
ϕ3,4 0 t4 + t8 + t12 t4 + t8 + t12 t2 + t6 + t10 t2 + t6 + t10 t2 t + t5 0 t3 t3 t + t5 t + t5 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k1,1 + k2,1 − 2k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ2,3'},   {ϕ2,5',  ϕ1,4,  ϕ3,2,  ϕ2,3''},   {ϕ2,5'',  ϕ2,7}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,4311 + 2t
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''322 + t
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2122 + 4t + 4t2 + 4t3 + 4t4 + 3t5
ϕ3,22133 + 4t + 4t2 + 4t3 + 4t4 + 2t5
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 0 1 1 1 1 1 0 0 0 1 0 1 1
ϕ1,4 1 1 1 1 1 1 1 0 1 0 0 0 0 1
ϕ1,8 1 0 1 1 1 1 0 0 0 1 1 1 0 1
ϕ1,6 1 0 1 1 1 1 0 0 1 1 0 1 0 1
ϕ1,10 1 0 1 1 1 1 0 0 0 1 1 0 1 1
ϕ1,14 1 0 1 1 1 1 1 1 1 0 0 0 0 1
ϕ2,5'' 2 0 2 2 2 2 2 0 2 0 0 1 1 2
ϕ2,3'' 2 0 2 2 2 2 2 1 1 0 1 0 1 2
ϕ2,3' 2 1 2 2 2 2 1 0 2 1 0 1 0 2
ϕ2,7 2 0 2 2 2 2 0 0 0 2 2 1 1 2
ϕ2,1 2 0 2 2 2 2 1 0 0 1 2 0 2 2
ϕ2,5' 2 0 2 2 2 2 0 0 1 2 1 2 0 2
ϕ3,2 3 0 3 3 3 3 1 0 2 2 1 1 2 3
ϕ3,4 3 0 3 3 3 3 2 0 1 1 2 2 1 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 0 t8 t6 t10 t14 t5 0 0 0 t 0 t2 t4
ϕ1,4 t8 1 t4 t14 t6 t10 t 0 t3 0 0 0 0 t4
ϕ1,8 t4 0 1 t10 t14 t6 0 0 0 t3 t5 t 0 t4
ϕ1,6 t6 0 t14 1 t4 t8 0 0 t t5 0 t3 0 t2
ϕ1,10 t14 0 t10 t8 1 t4 0 0 0 t t3 0 t4 t2
ϕ1,14 t10 0 t6 t4 t8 1 t3 t t5 0 0 0 0 t2
ϕ2,5'' t7 + t11 0 t3 + t7 t5 + t13 t5 + t9 t + t9 1 + t4 0 t2 + t6 0 0 t4 t 2t3
ϕ2,3'' t + t9 0 t5 + t9 t3 + t7 t7 + t11 t3 + t11 t2 + t6 1 t4 0 t2 0 t3 t + t5
ϕ2,3' t5 + t9 t t5 + t13 t3 + t11 t3 + t7 t7 + t11 t2 0 1 + t4 t4 0 t2 0 t + t5
ϕ2,7 t5 + t13 0 t + t9 t7 + t11 t3 + t11 t3 + t7 0 0 0 1 + t4 t2 + t6 t2 t3 t + t5
ϕ2,1 t3 + t11 0 t7 + t11 t5 + t9 t + t9 t5 + t13 t4 0 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 0 t3 + t11 t + t9 t5 + t13 t5 + t9 0 0 t2 t2 + t6 t4 1 + t4 0 2t3
ϕ3,2 t2 + t6 + t10 0 t2 + t6 + t10 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 t3 0 t + t5 t + t5 t3 t3 1 + t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 0 t4 + t8 + t12 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 t + t5 0 t3 t3 t + t5 t + t5 t2 1 + 2t4

For the generic point of the hyperplane k1,1 + k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ1,4,  ϕ1,14,  ϕ2,5'',  ϕ2,7},   {ϕ2,5',  ϕ2,3''},   {ϕ2,1,  ϕ2,3'}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,4911 + 2t + 3t2 + 2t3 + t4
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,14111
ϕ2,5''742 + 3t + 2t2
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 1 1 0 0 1 0 0 1 0 1 1
ϕ1,4 1 1 1 1 1 0 0 0 1 0 0 1 1 1
ϕ1,8 1 0 1 1 1 0 0 0 0 1 1 1 1 1
ϕ1,6 1 0 1 1 1 0 0 0 1 1 0 1 1 1
ϕ1,10 1 0 1 1 1 0 0 1 0 1 1 0 1 1
ϕ1,14 1 0 1 1 1 1 0 1 1 0 0 0 1 1
ϕ2,5'' 2 0 2 2 2 0 1 1 2 0 0 1 2 2
ϕ2,3'' 2 0 2 2 2 0 1 2 1 0 1 0 2 2
ϕ2,3' 2 1 2 2 2 0 0 0 2 1 0 2 2 2
ϕ2,7 2 0 2 2 2 0 0 1 0 2 2 1 2 2
ϕ2,1 2 1 2 2 2 0 0 2 0 1 2 0 2 2
ϕ2,5' 2 0 2 2 2 0 0 0 1 2 1 2 2 2
ϕ3,2 3 1 3 3 3 0 0 2 2 2 1 1 3 3
ϕ3,4 3 0 3 3 3 0 1 1 1 1 2 2 3 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 t6 t10 0 0 t3 0 0 t 0 t2 t4
ϕ1,4 t8 1 t4 t14 t6 0 0 0 t3 0 0 t5 t2 t4
ϕ1,8 t4 0 1 t10 t14 0 0 0 0 t3 t5 t t2 t4
ϕ1,6 t6 0 t14 1 t4 0 0 0 t t5 0 t3 t4 t2
ϕ1,10 t14 0 t10 t8 1 0 0 t5 0 t t3 0 t4 t2
ϕ1,14 t10 0 t6 t4 t8 1 0 t t5 0 0 0 t4 t2
ϕ2,5'' t7 + t11 0 t3 + t7 t5 + t13 t5 + t9 0 1 t2 t2 + t6 0 0 t4 t + t5 2t3
ϕ2,3'' t + t9 0 t5 + t9 t3 + t7 t7 + t11 0 t2 1 + t4 t4 0 t2 0 2t3 t + t5
ϕ2,3' t5 + t9 t t5 + t13 t3 + t11 t3 + t7 0 0 0 1 + t4 t4 0 t2 + t6 2t3 t + t5
ϕ2,7 t5 + t13 0 t + t9 t7 + t11 t3 + t11 0 0 t4 0 1 + t4 t2 + t6 t2 2t3 t + t5
ϕ2,1 t3 + t11 t3 t7 + t11 t5 + t9 t + t9 0 0 t2 + t6 0 t2 1 + t4 0 t + t5 2t3
ϕ2,5' t3 + t7 0 t3 + t11 t + t9 t5 + t13 0 0 0 t2 t2 + t6 t4 1 + t4 t + t5 2t3
ϕ3,2 t2 + t6 + t10 t2 t2 + t6 + t10 t4 + t8 + t12 t4 + t8 + t12 0 0 t + t5 t + t5 t + t5 t3 t3 1 + 2t4 2t2 + t6
ϕ3,4 t4 + t8 + t12 0 t4 + t8 + t12 t2 + t6 + t10 t2 + t6 + t10 0 t t3 t3 t3 t + t5 t + t5 2t2 + t6 1 + 2t4

For the generic point of the hyperplane k1,1 + k2,1 + k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ2,3'},   {ϕ1,0,  ϕ3,2,  ϕ2,5'',  ϕ2,7},   {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)
ϕ1,0311 + 2t
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,144811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ2,5''2122 + 4t + 4t2 + 4t3 + 4t4 + 3t5
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,7322 + t
ϕ2,12422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,22133 + 4t + 4t2 + 4t3 + 4t4 + 2t5
ϕ3,44833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 1 1 1 0 1 0 0 1 0 0 1
ϕ1,4 0 1 1 1 1 1 1 0 1 0 0 1 0 1
ϕ1,8 0 1 1 1 1 1 0 0 0 0 1 1 1 1
ϕ1,6 0 1 1 1 1 1 0 0 1 0 0 1 1 1
ϕ1,10 0 1 1 1 1 1 0 1 0 1 1 0 0 1
ϕ1,14 0 1 1 1 1 1 1 1 1 0 0 0 0 1
ϕ2,5'' 0 2 2 2 2 2 2 1 2 0 0 1 0 2
ϕ2,3'' 1 2 2 2 2 2 1 2 1 0 1 0 0 2
ϕ2,3' 0 2 2 2 2 2 1 0 2 0 0 2 1 2
ϕ2,7 0 2 2 2 2 2 0 1 0 1 2 1 1 2
ϕ2,1 0 2 2 2 2 2 1 2 0 0 2 0 1 2
ϕ2,5' 0 2 2 2 2 2 0 0 1 0 1 2 2 2
ϕ3,2 0 3 3 3 3 3 1 2 2 0 1 1 2 3
ϕ3,4 0 3 3 3 3 3 2 1 1 0 2 2 1 3

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 t6 t10 t14 0 t3 0 0 t 0 0 t4
ϕ1,4 0 1 t4 t14 t6 t10 t 0 t3 0 0 t5 0 t4
ϕ1,8 0 t8 1 t10 t14 t6 0 0 0 0 t5 t t2 t4
ϕ1,6 0 t10 t14 1 t4 t8 0 0 t 0 0 t3 t4 t2
ϕ1,10 0 t6 t10 t8 1 t4 0 t5 0 t t3 0 0 t2
ϕ1,14 0 t14 t6 t4 t8 1 t3 t t5 0 0 0 0 t2
ϕ2,5'' 0 t3 + t11 t3 + t7 t5 + t13 t5 + t9 t + t9 1 + t4 t2 t2 + t6 0 0 t4 0 2t3
ϕ2,3'' t t5 + t13 t5 + t9 t3 + t7 t7 + t11 t3 + t11 t2 1 + t4 t4 0 t2 0 0 t + t5
ϕ2,3' 0 t + t9 t5 + t13 t3 + t11 t3 + t7 t7 + t11 t2 0 1 + t4 0 0 t2 + t6 t3 t + t5
ϕ2,7 0 t5 + t9 t + t9 t7 + t11 t3 + t11 t3 + t7 0 t4 0 1 t2 + t6 t2 t3 t + t5
ϕ2,1 0 t3 + t7 t7 + t11 t5 + t9 t + t9 t5 + t13 t4 t2 + t6 0 0 1 + t4 0 t 2t3
ϕ2,5' 0 t7 + t11 t3 + t11 t + t9 t5 + t13 t5 + t9 0 0 t2 0 t4 1 + t4 t + t5 2t3
ϕ3,2 0 t2 + t6 + t10 t2 + t6 + t10 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 t3 t + t5 t + t5 0 t3 t3 1 + t4 2t2 + t6
ϕ3,4 0 t4 + t8 + t12 t4 + t8 + t12 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 t + t5 t3 t3 0 t + t5 t + t5 t2 1 + 2t4

For the generic point of the hyperplane k1,1 + 2k2,1 − k2,2

Quick navigation: Exceptional hyperplanes, For generic parameters

Non-singleton Calogero–Moser families

{ϕ2,1,  ϕ3,4,  ϕ1,14,  ϕ2,3'},   {ϕ2,5',  ϕ2,3''},   {ϕ2,5'',  ϕ2,7}

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 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,44811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,84811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,64811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,104811 + 2t + 3t2 + 4t3 + 4t4 + 4t5 + 4t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 3t12 + 2t13 + t14
ϕ1,14311 + 2t
ϕ2,5''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3''2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,3'2122 + 4t + 4t2 + 4t3 + 4t4 + 3t5
ϕ2,72422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ2,1322 + t
ϕ2,5'2422 + 4t + 4t2 + 4t3 + 4t4 + 4t5 + 2t6
ϕ3,24833 + 6t + 9t2 + 12t3 + 9t4 + 6t5 + 3t6
ϕ3,42133 + 4t + 4t2 + 4t3 + 4t4 + 2t5

Characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 1 1 1 1 0 1 1 0 0 1 0 1 0
ϕ1,4 1 1 1 1 1 0 1 0 1 0 0 1 1 0
ϕ1,8 1 1 1 1 1 0 0 0 0 1 0 1 1 1
ϕ1,6 1 1 1 1 1 0 0 0 1 1 0 1 1 0
ϕ1,10 1 1 1 1 1 0 0 1 0 1 0 0 1 1
ϕ1,14 1 1 1 1 1 1 1 1 0 0 0 0 1 0
ϕ2,5'' 2 2 2 2 2 1 2 1 1 0 0 1 2 0
ϕ2,3'' 2 2 2 2 2 0 2 2 1 0 0 0 2 1
ϕ2,3' 2 2 2 2 2 0 1 0 2 1 0 2 2 0
ϕ2,7 2 2 2 2 2 0 0 1 0 2 0 1 2 2
ϕ2,1 2 2 2 2 2 0 1 2 0 1 1 0 2 1
ϕ2,5' 2 2 2 2 2 0 0 0 1 2 0 2 2 1
ϕ3,2 3 3 3 3 3 0 1 2 2 2 0 1 3 1
ϕ3,4 3 3 3 3 3 0 2 1 1 1 0 2 3 2

Graded characters of the simple modules

ϕ L(ϕ1,0) L(ϕ1,4) L(ϕ1,8) L(ϕ1,6) L(ϕ1,10) L(ϕ1,14) L(ϕ2,5'') L(ϕ2,3'') L(ϕ2,3') L(ϕ2,7) L(ϕ2,1) L(ϕ2,5') L(ϕ3,2) L(ϕ3,4)
ϕ1,0 1 t4 t8 t6 t10 0 t5 t3 0 0 t 0 t2 0
ϕ1,4 t8 1 t4 t14 t6 0 t 0 t3 0 0 t5 t2 0
ϕ1,8 t4 t8 1 t10 t14 0 0 0 0 t3 0 t t2 t4
ϕ1,6 t6 t10 t14 1 t4 0 0 0 t t5 0 t3 t4 0
ϕ1,10 t14 t6 t10 t8 1 0 0 t5 0 t 0 0 t4 t2
ϕ1,14 t10 t14 t6 t4 t8 1 t3 t 0 0 0 0 t4 0
ϕ2,5'' t7 + t11 t3 + t11 t3 + t7 t5 + t13 t5 + t9 t 1 + t4 t2 t2 0 0 t4 t + t5 0
ϕ2,3'' t + t9 t5 + t13 t5 + t9 t3 + t7 t7 + t11 0 t2 + t6 1 + t4 t4 0 0 0 2t3 t
ϕ2,3' t5 + t9 t + t9 t5 + t13 t3 + t11 t3 + t7 0 t2 0 1 + t4 t4 0 t2 + t6 2t3 0
ϕ2,7 t5 + t13 t5 + t9 t + t9 t7 + t11 t3 + t11 0 0 t4 0 1 + t4 0 t2 2t3 t + t5
ϕ2,1 t3 + t11 t3 + t7 t7 + t11 t5 + t9 t + t9 0 t4 t2 + t6 0 t2 1 0 t + t5 t3
ϕ2,5' t3 + t7 t7 + t11 t3 + t11 t + t9 t5 + t13 0 0 0 t2 t2 + t6 0 1 + t4 t + t5 t3
ϕ3,2 t2 + t6 + t10 t2 + t6 + t10 t2 + t6 + t10 t4 + t8 + t12 t4 + t8 + t12 0 t3 t + t5 t + t5 t + t5 0 t3 1 + 2t4 t2
ϕ3,4 t4 + t8 + t12 t4 + t8 + t12 t4 + t8 + t12 t2 + t6 + t10 t2 + t6 + t10 0 t + t5 t3 t3 t3 0 t + t5 2t2 + t6 1 + t4