Table 17 Weyl group \(D_{4}\), level 7, 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, −4, 2 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ 1 & -1 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 15 \\ (25) \end{array}\) | 2, −3, 4, −2 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ 1 & -1 & -1 & 1 \\ 1 & -1 & 0 & 1 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 16 \\ (21) \end{array}\) | 1, 2, −5, −1 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & -1 & 0 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 17 \\ (17) \end{array}\) | −3, 5, −3, −3 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ -1 & 2 & -1 & -1 \\ -1 & 1 & 0 & -1 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 18 \\ (13) \end{array}\) | −1, −2, −1, 5 | \(\begin{array}{c} s_{3}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ 0 & 1 & 0 & -2 \\ -1 & 1 & 0 & -1 \\ 0 & 1 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{3}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 19 \\ (24) \end{array}\) | −1, 2, 1, −5 | \(\begin{array}{c} s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ 0 & 1 & 0 & -2 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{3}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 20 \\ (6) \end{array}\) | 1, −5, 3, 3 | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ 1 & 0 & -1 & -1 \\ 1 & 0 & 0 & -1 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3}s_{1}s_{2} \end{array}\) |
\(\begin{array}{c} 21 \\ (16) \end{array}\) | −1, 2, −5, 1 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & -1 & 0 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 22 \\ (27) \end{array}\) | −1, −2, 5, −1 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ 0 & 1 & -2 & 0 \\ 0 & 1 & -1 & 0 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 23 \\ (8) \end{array}\) | 3, −5, 1, 3 | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ -1 & 0 & 1 & -1 \\ -1 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 24 \\ (19) \end{array}\) | 1, 2, −1, −5 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{3}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ 0 & 1 & 0 & -2 \\ -1 & 1 & 0 & -1 \\ 0 & 0 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 25 \\ (15) \end{array}\) | 2, −3, −2, 4 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ 1 & -1 & 1 & -1 \\ 0 & -1 & 1 & 0 \\ 1 & -1 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 26 \\ (26) \end{array}\) | 2, −1, 2, −4 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 27 \\ (22) \end{array}\) | −1, 4, −5, −1 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1}s_{2}s_{4} \end{array}\) | \( \begin{bmatrix} 0 & 0 & 1 & -1 \\ 0 & -1 & 2 & 0 \\ 0 & -1 & 1 & 0 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3} \end{array}\) |