Table 9 Weyl group \(D_{4}\), level 4, elements 0–13
From: Extending Snow’s algorithm for computations in the finite Weyl groups
\(N^{o}\) | Weight | Element | Matrix | Inverse |
|---|---|---|---|---|
\(\begin{array}{c} 0 \\ (16) \end{array}\) | 1, 4, −3, −3 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 0 & 1 & 1 \\ -1 & 0 & 0 & 1 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 1 \\ (13) \end{array}\) | 2, −1, −2, 4 | \(\begin{array}{c} s_{3}s_{2}s_{3}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 1 & -1 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 2 \\ (20) \end{array}\) | 2, 1, 2, −4 | \(\begin{array}{c} s_{4}s_{2}s_{3}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 1 & -1 & 1 \\ 0 & 1 & -1 & 0 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 3 \\ (9) \end{array}\) | 3, −4, 3, 3 | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 2 & -1 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 4 \\ (14) \end{array}\) | 2, 1, −4, 2 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 1 & 1 & -1 \\ -1 & 1 & 0 & -1 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 5 \\ (21) \end{array}\) | 2, −1, 4, −2 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 1 & 0 & 0 \\ -1 & 1 & 1 & -1 \\ 0 & 0 & 1 & 0 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 6 \\ (12) \end{array}\) | −1, 2, −3, 3 | \(\begin{array}{c} s_{3}s_{2}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ -1 & 0 & 1 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 7 \\ (19) \end{array}\) | −1, 2, 3, −3 | \(\begin{array}{c} s_{4}s_{2}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ -1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 8 \\ (8) \end{array}\) | 1, −3, 1, 5 | \(\begin{array}{c} s_{2}s_{3}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 0 & -1 & 0 & 2 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{3}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 9 \\ (3) \end{array}\) | −2, 5, −2, −2 | \(\begin{array}{c} s_{4}s_{3}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ 1 & -1 & 0 & 1 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{1} \end{array}\) |
\(\begin{array}{c} 10 \\ (10) \end{array}\) | 1, −3, 5, 1 | \(\begin{array}{c} s_{2}s_{4}s_{1}s_{2} \end{array}\) | \( \begin{bmatrix} 0 & -1 & 1 & 1 \\ 0 & -1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 11 \\ (11) \end{array}\) | 5, −3, 1, 1 | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 1 & -1 & 0 & 1 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 12 \\ (6) \end{array}\) | −2, −1, 2, 4 | \(\begin{array}{c} s_{2}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ -1 & 1 & -1 & 1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 13 \\ (1) \end{array}\) | −3, 2, −1, 3 | \(\begin{array}{c} s_{3}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ 1 & 0 & -1 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{3}s_{1} \end{array}\) |