Tag 0042
\(D_{4}^{*}\)
D4*
Invariants
| Rank | 4 |
|---|---|
| Signature \((n_+, n_-)\) | (4, 0) |
| Determinant | \(\tfrac{1}{4}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{4}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | |
|---|---|---|---|---|
| \(e_{1}\) | 1 | 1 | 1/2 | 1/2 |
| \(e_{2}\) | 1 | 2 | 1 | 1 |
| \(e_{3}\) | 1/2 | 1 | 1 | 1/2 |
| \(e_{4}\) | 1/2 | 1 | 1/2 | 1 |
[[1, 1, 1/2, 1/2], [1, 2, 1, 1], [1/2, 1, 1, 1/2], [1/2, 1, 1/2, 1]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(1\) |
|---|---|
| Kissing number | 24 |
| Order of the isometry group | \(1152 = 2^{7} \cdot 3^{2}\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{1\}\) |
| Number of roots | 48 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong D_{4}(\tfrac{1}{2})\) |
| Rank of \(R(L)\) | 4 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(F_{4}\) | \(\{1, 2\}\) | 48 | \(\cong D_{4}(\tfrac{1}{2})\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} - 2e_{2} + e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{2} - 2e_{3}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{3} - e_{4}\) | \(1\) |
| \(\alpha_{4}\) | \(e_{4}\) | \(1\) |