Table 6 Weyl group \(D_{4}\), level 2, elements 0–8
From: Extending Snow’s algorithm for computations in the finite Weyl groups
\(N^{o}\) | Weight | Elem | Matrix | \(N^{o}\) | Weight | Elem | Matrix |
|---|---|---|---|---|---|---|---|
\(\begin{array}{c} 3 \\ (0)^{\mathrm{a}} \end{array} \) | −2, 1, 2, 2 | \(s_{1}s_{2}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} 6 \\ (4) \end{array} \) | 3, −2, 1, 3 | \(s_{2}s_{3}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) |
\(\begin{array}{c} 1 \\ (1) \end{array} \) | −1, 3, −1, 1 | \(s_{1}s_{3}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} 8 \\ (5) \end{array} \) | 3, −2, 3, 1 | \(s_{2}s_{4}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix} \) |
\(\begin{array}{c} 2 \\ (2) \end{array} \) | −1, 3, 1, −1 | \(s_{1}s_{4}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} 0 \\ (3) \end{array} \) | 1, −2, 3, 3 | \(s_{2}s_{1}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) |
\(\begin{array}{c} 4 \\ (6) \end{array} \) | 2, 1, −2, 2 | \(s_{3}s_{2}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & -1 & 1 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} 5 \\ (8) \end{array} \) | 2, 1, 2, −2 | \(s_{4}s_{2}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & -1 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) |
\(\begin{array}{c} 7 \\ (7) \end{array} \) | 1, 3, −1, −1 | \(s_{3}s_{4}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) |