Table 2 Conjugacy classes in the Weyl groups \(D_{4}\), see Tables 26–37 and Table 3
From: Extending Snow’s algorithm for computations in the finite Weyl groups
\(N^{\circ}\) | Carter diagrama | Representative element | Elms | Root subset | Order | Signed cycle-typeb |
|---|---|---|---|---|---|---|
0 | – | e | 1 | ∅ | 1 | [1111] |
1 |
| \(s_{1}\) | 12 | \(A_{1}\) | 2 | [211] |
2 |
| \(s_{1}s_{2}\) | 32 | \(A_{2}\) | 3 | [31] |
3 |
| \(s_{1}s_{3}\) | 6 | \(2A_{1}\) | 2 | [22] |
4 |
| \(s_{1}s_{4}\) | 6 | \(2A_{1}\) | 2 | [22] |
5 |
| \(s_{3}s_{4}\) | 6 | \(D_{2}\) | 2 | [\(\bar{1}\bar{1}\)11] |
6 |
| \(s_{1}s_{2}s_{3}\) | 24 | \(A_{3}\) | 4 | [4] |
7 |
| \(s_{1}s_{2}s_{4}\) | 24 | \(A_{3}\) | 4 | [4] |
8 |
| \(s_{1}s_{3}s_{4}\) | 12 | \(3A_{1}\) | 2 | [2\(\bar{1}\bar{1}\)] |
9 |
| \(s_{3}s_{2}s_{4}\) | 24 | \(D_{3}\) | 4 | [\(\bar{2}\bar{1}\)1] |
10 |
| \(s_{1}s_{4}s_{2}s_{3}\) | 32 | \(D_{4}\) | 6 | [\(\bar{3}\bar{1}\)] |
11 |
| \(s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}\) | 12 | \(D_{4}(a_{1})\) | 4 | [\(\bar{2}\bar{2}\)] |
12 |
| \(s_{1}s_{2}s_{3}s_{4}s_{2}s_{1}s_{2}s_{3}s_{4}s_{2}s_{3}s_{4}\) | 1 | \(4A_{1}\) | 2 | [\(\bar{1}\bar{1}\bar{1}\bar{1}\)] |
