Tag 0114
\(A_{9}^{*}\)
A9*
Invariants
| Rank | 9 |
|---|---|
| Signature \((n_+, n_-)\) | (9, 0) |
| Determinant | \(\tfrac{1}{10}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{9}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | |
|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 9/10 | 4/5 | 7/10 | 3/5 | 1/2 | 2/5 | 3/10 | 1/5 | 1/10 |
| \(e_{2}\) | 4/5 | 8/5 | 7/5 | 6/5 | 1 | 4/5 | 3/5 | 2/5 | 1/5 |
| \(e_{3}\) | 7/10 | 7/5 | 21/10 | 9/5 | 3/2 | 6/5 | 9/10 | 3/5 | 3/10 |
| \(e_{4}\) | 3/5 | 6/5 | 9/5 | 12/5 | 2 | 8/5 | 6/5 | 4/5 | 2/5 |
| \(e_{5}\) | 1/2 | 1 | 3/2 | 2 | 5/2 | 2 | 3/2 | 1 | 1/2 |
| \(e_{6}\) | 2/5 | 4/5 | 6/5 | 8/5 | 2 | 12/5 | 9/5 | 6/5 | 3/5 |
| \(e_{7}\) | 3/10 | 3/5 | 9/10 | 6/5 | 3/2 | 9/5 | 21/10 | 7/5 | 7/10 |
| \(e_{8}\) | 1/5 | 2/5 | 3/5 | 4/5 | 1 | 6/5 | 7/5 | 8/5 | 4/5 |
| \(e_{9}\) | 1/10 | 1/5 | 3/10 | 2/5 | 1/2 | 3/5 | 7/10 | 4/5 | 9/10 |
[[9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10], [4/5, 8/5, 7/5, 6/5, 1, 4/5, 3/5, 2/5, 1/5], [7/10, 7/5, 21/10, 9/5, 3/2, 6/5, 9/10, 3/5, 3/10], [3/5, 6/5, 9/5, 12/5, 2, 8/5, 6/5, 4/5, 2/5], [1/2, 1, 3/2, 2, 5/2, 2, 3/2, 1, 1/2], [2/5, 4/5, 6/5, 8/5, 2, 12/5, 9/5, 6/5, 3/5], [3/10, 3/5, 9/10, 6/5, 3/2, 9/5, 21/10, 7/5, 7/10], [1/5, 2/5, 3/5, 4/5, 1, 6/5, 7/5, 8/5, 4/5], [1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{9}{10}\) |
|---|---|
| Kissing number | 20 |
| Order of the isometry group | \(7257600 = 2^{9} \cdot 3^{4} \cdot 5^{2} \cdot 7\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 90 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{9}\) |
| Rank of \(R(L)\) | 9 |
| \([L : R(L)]\) | 10 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/10\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{9}\) | \(\{2\}\) | 90 | \(\cong A_{9}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{9}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{8} - 2e_{9}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{7} - 2e_{8} + e_{9}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{6} - 2e_{7} + e_{8}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{5} - 2e_{6} + e_{7}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |