Tag 0054
\(A_{5}^{*}\)
A5*
Invariants
| Rank | 5 |
|---|---|
| Signature \((n_+, n_-)\) | (5, 0) |
| Determinant | \(\tfrac{1}{6}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{5}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | |
|---|---|---|---|---|---|
| \(e_{1}\) | 5/6 | 2/3 | 1/2 | 1/3 | 1/6 |
| \(e_{2}\) | 2/3 | 4/3 | 1 | 2/3 | 1/3 |
| \(e_{3}\) | 1/2 | 1 | 3/2 | 1 | 1/2 |
| \(e_{4}\) | 1/3 | 2/3 | 1 | 4/3 | 2/3 |
| \(e_{5}\) | 1/6 | 1/3 | 1/2 | 2/3 | 5/6 |
[[5/6, 2/3, 1/2, 1/3, 1/6], [2/3, 4/3, 1, 2/3, 1/3], [1/2, 1, 3/2, 1, 1/2], [1/3, 2/3, 1, 4/3, 2/3], [1/6, 1/3, 1/2, 2/3, 5/6]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{5}{6}\) |
|---|---|
| Kissing number | 12 |
| Order of the isometry group | \(1440 = 2^{5} \cdot 3^{2} \cdot 5\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 30 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{5}\) |
| Rank of \(R(L)\) | 5 |
| \([L : R(L)]\) | 6 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/6\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{5}\) | \(\{2\}\) | 30 | \(\cong A_{5}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{5}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{4} - 2e_{5}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |