atlas Library · Script File
springer_table_E7.at
atlas> show_nilpotent_orbits("E7".simply_connected)
i H diagram dim BC Levi Cent A(O)
0 [0,0,0,0,0,0,0] [0,0,0,0,0,0,0] 0 7T1 E7 [1]
1 [2,2,3,4,3,2,1] [1,0,0,0,0,0,0] 34 A1+6T1 D6 [1]
2 [2,3,4,6,5,4,2] [0,0,0,0,0,1,0] 52 2A1+5T1 A1+B4 [1]
3 [2,3,4,6,5,4,3] [0,0,0,0,0,0,2] 54 3A1+4T1 F4 [1,2]
4 [3,4,6,8,6,4,2] [0,0,1,0,0,0,0] 64 3A1+4T1 A1+C3 [1]
5 [4,4,6,8,6,4,2] [2,0,0,0,0,0,0] 66 A2+5T1 A5 [1,2]
6 [3,5,6,9,7,5,3] [0,1,0,0,0,0,1] 70 4A1+3T1 C3 [1,2]
7 [4,5,7,10,8,6,3] [1,0,0,0,0,1,0] 76 A1+A2+4T1 A3+T1 [1,2]
8 [4,6,8,12,9,6,3] [0,0,0,1,0,0,0] 82 2A1+A2+3T1 3A1 [1]
9 [6,7,10,14,11,8,4] [2,0,0,0,0,1,0] 84 A3+4T1 A1+B3 [1]
10 [4,6,8,12,10,8,4] [0,0,0,0,0,2,0] 84 2A2+3T1 A1+G2 [1]
11 [4,7,8,12,9,6,3] [0,2,0,0,0,0,0] 84 3A1+A2+2T1 G2 [1,2]
12 [6,7,10,14,11,8,5] [2,0,0,0,0,0,2] 86 A1+A3+3T1 B3 [1,2]
13 [5,7,10,14,11,8,4] [0,0,1,0,0,1,0] 90 A1+2A2+2T1 2A1 [1]
14 [6,8,11,16,12,8,4] [1,0,0,1,0,0,0] 92 A1+A3+3T1 3A1 [1]
15 [6,8,12,16,12,8,4] [0,0,2,0,0,0,0] 94 D4+3T1 3A1 [1,2,3]
16 [6,8,11,16,13,9,5] [1,0,0,0,1,0,1] 94 2A1+A3+2T1 2A1 [1,2]
17 [10,12,18,24,18,12,6] [2,0,2,0,0,0,0] 96 D4+3T1 C3 [1]
18 [6,9,12,17,13,9,5] [0,1,1,0,0,0,1] 96 A1+D4+2T1 2A1 [1,2,2,2]
19 [6,9,12,18,14,10,5] [0,0,0,1,0,1,0] 98 A2+A3+2T1 A1+T1 [1,2]
20 [8,10,14,20,16,12,6] [2,0,0,0,0,2,0] 100 A4+3T1 A2+T1 [1,2]
21 [6,9,12,18,15,10,5] [0,0,0,0,2,0,0] 100 A1+A2+A3+T1 A1 [1,2]
22 [10,13,18,25,19,13,7] [2,1,1,0,0,0,1] 102 A1+D4+2T1 B2 [1,2]
23 [10,13,18,26,21,16,9] [2,0,0,0,0,2,2] 102 A5+2T1 G2 [1,2]
24 [8,11,15,22,17,12,6] [1,0,0,1,0,1,0] 104 A1+A4+2T1 2T1 [1,2]
25 [10,13,18,26,20,14,7] [2,0,0,1,0,1,0] 106 D5+2T1 A1+T1 [1,2]
26 [8,12,16,24,18,12,6] [0,0,0,2,0,0,0] 106 A2+A4+T1 A1 [1]
27 [10,14,19,28,22,16,8] [1,0,0,1,0,2,0] 108 A5+2T1 2A1 [1]
28 [10,13,18,26,21,14,7] [2,0,0,0,2,0,0] 108 A1+D5+T1 A1 [1,2]
29 [10,14,19,28,22,16,9] [1,0,0,1,0,1,2] 108 A1+A5+T1 A1 [1,2]
30 [10,14,20,28,22,16,8] [0,0,2,0,0,2,0] 110 E6+T1 A1 [1,2]
31 [10,15,20,29,23,16,9] [0,1,1,0,1,0,2] 110 D6+T1 A1 [1,2]
32 [14,18,26,36,28,20,10] [2,0,2,0,0,2,0] 112 D5+2T1 2A1 [1]
33 [10,15,20,30,23,16,9] [0,0,0,2,0,0,2] 112 E7 e [1,2,2,2,3,6]
34 [14,19,26,37,29,20,10] [2,1,1,0,1,1,0] 114 A1+D5+T1 A1 [1,2]
35 [12,18,24,36,28,20,10] [0,0,0,2,0,2,0] 114 A6+T1 A1 [1]
36 [14,19,26,37,29,20,11] [2,1,1,0,1,0,2] 114 D6+T1 A1 [1,2]
37 [14,19,26,38,29,20,11] [2,0,0,2,0,0,2] 116 E7 e [1,2,2,2]
38 [16,22,30,44,34,24,12] [2,0,0,2,0,2,0] 118 E6+T1 T1 [1,2]
39 [18,25,34,49,39,28,15] [2,1,1,0,1,2,2] 118 D6+T1 A1 [1,2]
40 [22,30,42,60,46,32,16] [2,0,2,2,0,2,0] 120 E6+T1 A1 [1]
41 [18,25,34,50,39,28,15] [2,0,0,2,0,2,2] 120 E7 e [1,2,2,2]
42 [22,31,42,60,47,32,17] [2,2,2,0,2,0,2] 122 E7 e [1,2]
43 [26,37,50,72,57,40,21] [2,2,2,0,2,2,2] 124 E7 e [1,2]
44 [34,49,66,96,75,52,27] [2,2,2,2,2,2,2] 126 E7 e [1,2]
The duality on these orbits is given as follows:
Name Diagram Dim Spec Dual A:Abar Dual Diagram
0 0000000 0 1 E7 1 2222222
A1 1000000 34 1 E7(a1) 1 2220222
2A1 0000010 52 1 E7(a2) 1 2220202
(3A1)" 0000002 54 1 E6 Z2 2022020
(3A1)' 0010000 64 0 E7(a3) 1 2002022
A2 2000000 66 1 E7(a3) S2 2002022
4A1 0100001 70 0 E6(a1) 1 2002020
A2+A1 1000010 76 1 E6(a1) S2 2002020
A2+2A1 0001000 82 1 E7(a4) 1 2002002
A3 2000010 84 1 D6(a1) 1 2110102
2A2 0000020 84 1 D5+A1 1 2110110
A2+3A1 0200000 84 1 A6 Z2 0002020
(A3+A1)" 2000002 86 1 D5 Z2 2020020
2A2+A1 0010010 90 0 E7(a5) 1 0002002
(A3+A1)' 1001000 92 0 E7(a5) 1 0002002
D4(a1) 0020000 94 1 E7(a5) S3 0002002
A3+2A1 1000101 94 0 E6(a3) Z2 0020020
D4 2020000 96 1 (A5)" 1 2000022
D4(a1)+A1 0110001 96 1 E6(a3) S2Z2 0020020
A3+A2 0001010 98 1 D5(a1)+A1 S2:1 2000200
A4 2000020 100 1 D5(a1) S2 2001010
A3+A2+A1 0000200 100 1 A4+A2 Z2 0002000
(A5)" 2000022 102 1 D4 1 2020000
D4+A1 2110001 102 0 A4 Z2 2000020
A4+A1 1001010 104 1 A4+A1 S2 1001010
D5(a1) 2001010 106 1 A4 S2 2000020
A4+A2 0002000 106 1 A3+A2+A1 1 0000200
(A5)' 1001020 108 0 D4(a1)+A1 1 0110001
A5+A1 1001012 108 0 D4(a1) Z2 0020000
D5(a1)+A1 2000200 108 1 A3+A2 Z2 0001010
D6(a2) 0110102 110 0 D4(a1) Z2 0020000
E6(a3) 0020020 110 1 D4(a1)+A1 S2 0110001
D5 2020020 112 1 (A3+A1)" 1 2000002
E7(a5) 0002002 112 1 D4(a1) S3Z2 0020000
A6 0002020 114 1 A2+3A1 1 0200000
D5+A1 2110110 114 1 2A2 Z2 0000020
D6(a1) 2110102 114 1 A3 Z2 2000010
E7(a4) 2002002 116 1 A2+2A1 S2Z2:Z2 0001000
D6 2110122 118 0 A2 Z2 2000000
E6(a1) 2002020 118 1 A2+A1 S2 1000010
E6 2022020 120 1 (3A1)" 1 0000002
E7(a3) 2002022 120 1 A2 S2Z2 2000000
E7(a2) 2220202 122 1 2A1 Z2 0000010
E7(a1) 2220222 124 1 A1 Z2 1000000
E7 2222222 126 1 0 Z2 0000000Definitions in source order
E7_nilpotent_orbit_dual_map
L104
E7_nilpotent_orbit_dual_map = (vec->vec)E7_Springer_map
L166
E7_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 |e7_table| from character_table_E7,at
orbit diagrams as |vec| values
springer_table_E7 2 overloads
L224
springer_table_E7 () = SpringerTableL242
springer_table_E7 (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 |E7_nilpotent_orbit_dual_map| into the order of |dual(rd)|.
also in this file at line 370
dual_map_E7
L258
dual_map_E7(ComplexNilpotent O) = ComplexNilpotent[0,0,0,0,0,0,0] e 1 63 [1,0,0,0,0,0,0] A1 7,46 [0,0,0,0,0,1,0] 2A1 27,37 [0,0,0,0,0,0,2] 3A1 21,36 [0,0,1,0,0,0,0] 3A1 35,31 [2,0,0,0,0,0,0] A2 56,30 [0,1,0,0,0,0,1] 4A1 15,28 [1,0,0,0,0,1,0] A1+A2 120,25 [0,0,0,1,0,0,0] 2A1+A2 189,22 [2,0,0,0,0,1,0] A3 210,21 [0,0,0,0,0,2,0] 2A2 168,21 [0,2,0,0,0,0,0] 3A1+A2 105,21 [2,0,0,0,0,0,2] A1+A3 189,20 [0,0,1,0,0,1,0] A1+2A2 70,18 [1,0,0,1,0,0,0] A1+A3 280,17 [0,0,2,0,0,0,0] D4 315,16 [1,0,0,0,1,0,1] 2A1+A3 216,16 [2,0,2,0,0,0,0] D4 105,15 [0,1,1,0,0,0,1] A1+D4 405,15 [0,0,0,1,0,1,0] A2+A3 378,14 [2,0,0,0,0,2,0] A4 420,13 [0,0,0,0,2,0,0] A1+A2+A3 210,13 [2,1,1,0,0,0,1] A1+D4 84,12 [2,0,0,0,0,2,2] A5 105,12 [1,0,0,1,0,1,0] A1+A4 512,11 [2,0,0,1,0,1,0] D5 420,10 [0,0,0,2,0,0,0] A2+A4 210,10 [1,0,0,1,0,2,0] A5 216, 9 [2,0,0,0,2,0,0] A1+D5 378, 9 [1,0,0,1,0,1,2] A1+A5 70, 9 [0,0,2,0,0,2,0] E6 405, 8 [0,1,1,0,1,0,2] D6 280, 8 [2,0,2,0,0,2,0] D5 189, 7 [0,0,0,2,0,0,2] E7 315, 7 [2,1,1,0,1,1,0] A1+D5 168, 6 [0,0,0,2,0,2,0] A6 105, 6 [2,1,1,0,1,0,2] D6 210, 6 [2,0,0,2,0,0,2] E7 189, 5 [2,0,0,2,0,2,0] E6 120, 4 [2,1,1,0,1,2,2] D6 35, 4 [2,0,2,2,0,2,0] E6 21, 3 [2,0,0,2,0,2,2] E7 56, 3 [2,2,2,0,2,0,2] E7 27, 2 [2,2,2,0,2,2,2] E7 7, 1 [2,2,2,2,2,2,2] E7 1, 0L262
springer_by_signature_lookup_E7
L310
springer_by_signature_lookup_E7 = [ [int],(int,int) ]springer_by_signature_E7
L358
springer_by_signature_E7 (ComplexNilpotent O) = (int,int)springer_E7
L364
springer_E7 (CharacterTable ct,ComplexNilpotent O) = intspringer_table_E7
L370
springer_table_E7 (CharacterTable ct) = SpringerTableGenerated from atlas-scripts at commit 648506c (2026-10-13).