Table 12 Weyl group \(D_{4}\), level 5, elements 14–27
From: Extending Snow’s algorithm for computations in the finite Weyl groups
\(N^{o}\) | Weight | Element | Matrix | Inverse |
|---|---|---|---|---|
\(\begin{array}{c} 14 \\ (14) \end{array}\) | −2, 1, −2, 4 | \(\begin{array}{c} s_{3}s_{2}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ -1 & 1 & -1 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 15 \\ (23) \end{array}\) | −2, 3, 2, −4 | \(\begin{array}{c} s_{4}s_{2}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ -1 & 1 & -1 & 1 \\ 0 & 1 & -1 & 0 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 16 \\ (6) \end{array}\) | −1, −2, 1, 5 | \(\begin{array}{c} s_{2}s_{3}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ 0 & -1 & 0 & 2 \\ 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{3}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 17 \\ (2) \end{array}\) | −3, 5, −1, −3 | \(\begin{array}{c} s_{4}s_{3}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ 1 & 0 & -1 & 1 \\ 1 & -1 & 0 & 1 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1} \end{array}\) |
\(\begin{array}{c} 18 \\ (9) \end{array}\) | 1, −4, 5, 1 | \(\begin{array}{c} s_{2}s_{4}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ 0 & 1 & -2 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 19 \\ (5) \end{array}\) | −3, −1, 3, 3 | \(\begin{array}{c} s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ -1 & 2 & -1 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 20 \\ (1) \end{array}\) | −4, 3, −2, 2 | \(\begin{array}{c} s_{3}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{3}s_{1} \end{array}\) |
\(\begin{array}{c} 21 \\ (4) \end{array}\) | −4, 3, 2, −2 | \(\begin{array}{c} s_{4}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ 1 & 1 & -1 & -1 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1} \end{array}\) |
\(\begin{array}{c} 22 \\ (22) \end{array}\) | 4, 1, −2, −2 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & -1 & -1 \\ 1 & 0 & 0 & -1 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 23 \\ (15) \end{array}\) | −2, 3, −4, 2 | \(\begin{array}{c} s_{3}s_{2}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ -1 & 1 & 1 & -1 \\ -1 & 1 & 0 & -1 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 24 \\ (24) \end{array}\) | −2, 1, 4, −2 | \(\begin{array}{c} s_{4}s_{2}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ -1 & 1 & 1 & -1 \\ 0 & 0 & 1 & 0 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 25 \\ (7) \end{array}\) | 1, −4, 1, 5 | \(\begin{array}{c} s_{2}s_{3}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ 0 & 1 & 0 & -2 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{3}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 26 \\ (3) \end{array}\) | −3, 5, −3, −1 | \(\begin{array}{c} s_{4}s_{3}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ 1 & 0 & 1 & -1 \\ 1 & 0 & 0 & -1 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{1} \end{array}\) |
\(\begin{array}{c} 27 \\ (10) \end{array}\) | −1, −2, 5, 1 | \(\begin{array}{c} s_{2}s_{4}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ 0 & -1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{1}s_{2} \end{array}\) |