Table 13 Weyl group \(D_{4}\), level 6, elements 0–13
From: Extending Snow’s algorithm for computations in the finite Weyl groups
\(N^{o}\) | Weight | Element | Matrix | Inverse |
|---|---|---|---|---|
\(\begin{array}{c} 0 \\ (0) \end{array}\) | −5, 1, 1, 1 | \(\begin{array}{c} s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 0 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 1 \\ (11) \end{array}\) | 5, −3, −1, 1 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ -1 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 2 \\ (10) \end{array}\) | 5, −3, 1, −1 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 3 \\ (22) \end{array}\) | 3, 2, −3, −3 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 2 & -1 & -1 \\ -1 & 1 & 0 & -1 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 4 \\ (9) \end{array}\) | 4, −5, 2, 2 | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{2}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ -1 & -1 & 1 & 1 \\ -1 & 0 & 0 & 1 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 5 \\ (21) \end{array}\) | 1, 3, −1, −5 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{3}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 0 & -1 & 0 & 2 \\ -1 & 0 & 0 & 1 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{3}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 6 \\ (16) \end{array}\) | 3, −2, −3, 3 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 1 & -2 & 1 & 1 \\ 0 & -1 & 1 & 0 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 7 \\ (27) \end{array}\) | 3, −2, 3, −3 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 1 & -2 & 1 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 8 \\ (23) \end{array}\) | 1, 3, −5, −1 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 0 & -1 & 2 & 0 \\ 0 & -1 & 1 & 0 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 9 \\ (4) \end{array}\) | −3, −2, 3, 3 | \(\begin{array}{c} s_{2}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 1 & -2 & 1 & 1 \\ 1 & -1 & 0 & 1 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{2}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 10 \\ (2) \end{array}\) | −5, 3, −1, 1 | \(\begin{array}{c} s_{3}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 1 & 0 & 0 & -1 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 11 \\ (1) \end{array}\) | −5, 3, 1, −1 | \(\begin{array}{c} s_{4}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 12 \\ (12) \end{array}\) | 5, −1, −1, −1 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 1 & 0 & 0 & -1 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 13 \\ (19) \end{array}\) | −2, 5, −2, −4 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ -1 & 1 & -1 & 1 \\ -1 & 0 & 0 & 1 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) |