Table 20 Weyl group \(D_{4}\), level 9, elements 0–13
From: Extending Snow’s algorithm for computations in the finite Weyl groups
\(N^{o}\) | Weight | Element | Matrix | Inverse |
|---|---|---|---|---|
\(\begin{array}{c} 0 \\ (3) \end{array}\) | −4, 3, −2, −2 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 0 & 0 & 0 \\ -1 & -1 & 1 & 1 \\ 0 & -1 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 1 \\ (5) \end{array}\) | −3, −1, −1, 3 | \(\begin{array}{c} s_{3}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 0 & 0 & 0 \\ -1 & 0 & 1 & -1 \\ 0 & -1 & 1 & 0 \\ -1 & 0 & 1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 2 \\ (2) \end{array}\) | −3, 1, 1, −3 | \(\begin{array}{c} s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 0 & 0 & 0 \\ -1 & 0 & 1 & -1 \\ -1 & 1 & 0 & -1 \\ 0 & 0 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 3 \\ (0) \end{array}\) | −2, −3, 2, 2 | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 0 & 0 & 0 \\ -1 & 1 & -1 & -1 \\ -1 & 1 & 0 & -1 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 4 \\ (4) \end{array}\) | −3, 1, −3, 1 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 0 & 0 & 0 \\ -1 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 5 \\ (1) \end{array}\) | −3, −1, 3, −1 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) | \( \begin{bmatrix} -1 & 0 & 0 & 0 \\ -1 & 0 & -1 & 1 \\ -1 & 0 & 0 & 1 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2}s_{1} \end{array}\) |
\(\begin{array}{c} 6 \\ (6) \end{array}\) | 1, −4, 1, 1 | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 7 \\ (13) \end{array}\) | −2, 3, −2, −4 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 1 & -1 & 1 & -1 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 8 \\ (11) \end{array}\) | −1, −1, −3, 3 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 1 & 0 & -1 & -1 \\ 0 & 0 & -1 & 0 \\ 1 & 0 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{3} \end{array}\) |
\(\begin{array}{c} 9 \\ (15) \end{array}\) | −1, −1, 3, −3 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 1 & 0 & -1 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4} \end{array}\) |
\(\begin{array}{c} 10 \\ (12) \end{array}\) | −2, 3, −4, −2 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) | \( \begin{bmatrix} 1 & -1 & 0 & 0 \\ 1 & -1 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) |
\(\begin{array}{c} 11 \\ (8) \end{array}\) | 3, −1, −3, −1 | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 0 & -1 & 1 \\ -1 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 1 \end{bmatrix} \) | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 12 \\ (10) \end{array}\) | 2, −3, −2, 2 | \(\begin{array}{c} s_{3}s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ 0 & 0 & -1 & 0 \\ -1 & 1 & -1 & 0 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{4}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |
\(\begin{array}{c} 13 \\ (7) \end{array}\) | 2, −3, 2, −2 | \(\begin{array}{c} s_{4}s_{2}s_{4}s_{3}s_{2}s_{1}s_{2}s_{4}s_{3} \end{array}\) | \( \begin{bmatrix} 0 & 1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & 1 & 0 & -1 \\ 0 & 0 & 0 & -1 \end{bmatrix} \) | \(\begin{array}{c} s_{4}s_{3}s_{2}s_{3}s_{1}s_{2}s_{4}s_{3}s_{2} \end{array}\) |