atlas Library · Script File
springer_table_E6.at
Let are 21 nilpotent orbits for E6:
atlas> show_nilpotent_orbits("E6".simply_connected)
i H diagram dim BC Levi Cent A(O)
0 [0,0,0,0,0,0] [0,0,0,0,0,0] 0 6T1 E6 [1]
1 [1,2,2,3,2,1] [0,1,0,0,0,0] 22 A1+5T1 A5 [1]
2 [2,2,3,4,3,2] [1,0,0,0,0,1] 32 2A1+4T1 B3+T1 [1]
3 [2,3,4,6,4,2] [0,0,0,1,0,0] 40 3A1+3T1 A1+A2 [1]
4 [2,4,4,6,4,2] [0,2,0,0,0,0] 42 A2+4T1 2A2 [1,2]
5 [3,4,5,7,5,3] [1,1,0,0,0,1] 46 A1+A2+3T1 A2+T1 [1]
6 [4,4,6,8,6,4] [2,0,0,0,0,2] 48 2A2+2T1 G2 [1,3,3]
7 [3,4,6,8,6,3] [0,0,1,0,1,0] 50 2A1+A2+2T1 A1+T1 [1]
8 [4,6,7,10,7,4] [1,2,0,0,0,1] 52 A3+3T1 B2+T1 [1]
9 [4,5,7,10,7,4] [1,0,0,1,0,1] 54 A1+2A2+T1 A1 [1,3,3]
10 [4,6,8,11,8,4] [0,1,1,0,1,0] 56 A1+A3+2T1 A1+T1 [1]
11 [4,6,8,12,8,4] [0,0,0,2,0,0] 58 D4+2T1 e [1,2,3]
12 [6,8,10,14,10,6] [2,2,0,0,0,2] 60 A4+2T1 A1+T1 [1]
13 [6,10,12,18,12,6] [0,2,0,2,0,0] 60 D4+2T1 A2 [1]
14 [6,8,11,15,11,6] [1,1,1,0,1,1] 62 A1+A4+T1 e [1]
15 [7,10,13,18,13,7] [1,2,1,0,1,1] 64 D5+T1 e [1]
16 [8,10,14,19,14,8] [2,1,1,0,1,2] 64 A5+T1 A1 [1,3,3]
17 [8,10,14,20,14,8] [2,0,0,2,0,2] 66 E6 e [1,2,3,3,6,6]
18 [10,14,18,26,18,10] [2,2,0,2,0,2] 68 D5+T1 e [1]
19 [12,16,22,30,22,12] [2,2,2,0,2,2] 70 E6 e [1,3,3]
20 [16,22,30,42,30,16] [2,2,2,2,2,2] 72 E6 e [1,3,3]
The duality on these orbits is given as follows:
Name Diagram Dim Spec Dual A:Abar Dual_Diagram
0 000000 0 1 E6 1 222222
A1 010000 22 1 E6(a1) 1 222022
2A1 100001 32 1 D5 1 220202
3A1 000100 40 0 E6(a3) 1 200202
A2 020000 42 1 E6(a3) S2 200202
A2+A1 110001 46 1 D5(a1) 1 121011
2A2 200002 48 1 D4 Z3 020200
A2+2A1 001010 50 1 A4+A1 1 111011
A3 120001 52 1 A4 1 220002
2A2+A1 100101 54 0 D4(a1) Z3 000200
A3+A1 011010 56 0 D4(a1) 1 000200
D4(a1) 000200 58 1 D4(a1) S3 000200
A4 220002 60 1 A3 1 120001
D4 020200 60 1 2A2 1 200002
A4+A1 111011 62 1 A2+2A1 1 100101
A5 211012 64 0 A2 Z3 020000
D5(a1) 121011 64 1 A2+A1 1 110001
E6(a3) 200202 66 1 A2 S2Z3 020000
D5 220202 68 1 2A1 1 100001
E6(a1) 222022 70 1 A1 Z3 010000
E6 222222 72 1 0 Z3 000000Definitions in source order
E6_nilpotent_orbit_dual_map
L57
E6_nilpotent_orbit_dual_map = (vec->vec)E6_Springer_map
L95
E6_Springer_map = (vec->int)The "Springer map" is given by the following mapping in terms of orbit diagrams; the image value are character indices according to the numbering of |e6_table| from character_table_E6,at
orbit diagrams as |vec| values
springer_table_E6 2 overloads
L129
springer_table_E6 () = SpringerTableL147
springer_table_E6 (RootDatum rd) = SpringerTableIn the general case we take into account the permutation of the simple root indices returned by |test_simple_type| (it is a map from Bourbaki indices to |rd| indices, which |stratified_diagram@(ComplexNilpotent,[int])| takes into account to get into Bourbaki order, and which |permute| uses to transform the dual diagram from |E6_nilpotent_orbit_dual_map| into the order of |dual(rd)|.
Generated from atlas-scripts at commit 648506c (2026-10-13).