Tag 0100
\(A_{8}^{*}\)
A8*
Invariants
| Rank | 8 |
|---|---|
| Signature \((n_+, n_-)\) | (8, 0) |
| Determinant | \(\tfrac{1}{9}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{8}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | |
|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 8/9 | 7/9 | 2/3 | 5/9 | 4/9 | 1/3 | 2/9 | 1/9 |
| \(e_{2}\) | 7/9 | 14/9 | 4/3 | 10/9 | 8/9 | 2/3 | 4/9 | 2/9 |
| \(e_{3}\) | 2/3 | 4/3 | 2 | 5/3 | 4/3 | 1 | 2/3 | 1/3 |
| \(e_{4}\) | 5/9 | 10/9 | 5/3 | 20/9 | 16/9 | 4/3 | 8/9 | 4/9 |
| \(e_{5}\) | 4/9 | 8/9 | 4/3 | 16/9 | 20/9 | 5/3 | 10/9 | 5/9 |
| \(e_{6}\) | 1/3 | 2/3 | 1 | 4/3 | 5/3 | 2 | 4/3 | 2/3 |
| \(e_{7}\) | 2/9 | 4/9 | 2/3 | 8/9 | 10/9 | 4/3 | 14/9 | 7/9 |
| \(e_{8}\) | 1/9 | 2/9 | 1/3 | 4/9 | 5/9 | 2/3 | 7/9 | 8/9 |
[[8/9, 7/9, 2/3, 5/9, 4/9, 1/3, 2/9, 1/9], [7/9, 14/9, 4/3, 10/9, 8/9, 2/3, 4/9, 2/9], [2/3, 4/3, 2, 5/3, 4/3, 1, 2/3, 1/3], [5/9, 10/9, 5/3, 20/9, 16/9, 4/3, 8/9, 4/9], [4/9, 8/9, 4/3, 16/9, 20/9, 5/3, 10/9, 5/9], [1/3, 2/3, 1, 4/3, 5/3, 2, 4/3, 2/3], [2/9, 4/9, 2/3, 8/9, 10/9, 4/3, 14/9, 7/9], [1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{8}{9}\) |
|---|---|
| Kissing number | 18 |
| Order of the isometry group | \(725760 = 2^{8} \cdot 3^{4} \cdot 5 \cdot 7\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 72 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{8}\) |
| Rank of \(R(L)\) | 8 |
| \([L : R(L)]\) | 9 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/9\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{8}\) | \(\{2\}\) | 72 | \(\cong A_{8}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{8}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{7} - 2e_{8}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{6} - 2e_{7} + e_{8}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{5} - 2e_{6} + e_{7}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |