Tag 0068
\(A_{6}^{*}\)
A6*
Invariants
| Rank | 6 |
|---|---|
| Signature \((n_+, n_-)\) | (6, 0) |
| Determinant | \(\tfrac{1}{7}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{6}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | |
|---|---|---|---|---|---|---|
| \(e_{1}\) | 6/7 | 5/7 | 4/7 | 3/7 | 2/7 | 1/7 |
| \(e_{2}\) | 5/7 | 10/7 | 8/7 | 6/7 | 4/7 | 2/7 |
| \(e_{3}\) | 4/7 | 8/7 | 12/7 | 9/7 | 6/7 | 3/7 |
| \(e_{4}\) | 3/7 | 6/7 | 9/7 | 12/7 | 8/7 | 4/7 |
| \(e_{5}\) | 2/7 | 4/7 | 6/7 | 8/7 | 10/7 | 5/7 |
| \(e_{6}\) | 1/7 | 2/7 | 3/7 | 4/7 | 5/7 | 6/7 |
[[6/7, 5/7, 4/7, 3/7, 2/7, 1/7], [5/7, 10/7, 8/7, 6/7, 4/7, 2/7], [4/7, 8/7, 12/7, 9/7, 6/7, 3/7], [3/7, 6/7, 9/7, 12/7, 8/7, 4/7], [2/7, 4/7, 6/7, 8/7, 10/7, 5/7], [1/7, 2/7, 3/7, 4/7, 5/7, 6/7]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{6}{7}\) |
|---|---|
| Kissing number | 14 |
| Order of the isometry group | \(10080 = 2^{5} \cdot 3^{2} \cdot 5 \cdot 7\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 42 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{6}\) |
| Rank of \(R(L)\) | 6 |
| \([L : R(L)]\) | 7 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/7\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{6}\) | \(\{2\}\) | 42 | \(\cong A_{6}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{6}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{5} - 2e_{6}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |