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