Tag 0041
\(A_{4}^{*}\)
A4*
Invariants
| Rank | 4 |
|---|---|
| Signature \((n_+, n_-)\) | (4, 0) |
| Determinant | \(\tfrac{1}{5}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{4}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | |
|---|---|---|---|---|
| \(e_{1}\) | 4/5 | 3/5 | 2/5 | 1/5 |
| \(e_{2}\) | 3/5 | 6/5 | 4/5 | 2/5 |
| \(e_{3}\) | 2/5 | 4/5 | 6/5 | 3/5 |
| \(e_{4}\) | 1/5 | 2/5 | 3/5 | 4/5 |
[[4/5, 3/5, 2/5, 1/5], [3/5, 6/5, 4/5, 2/5], [2/5, 4/5, 6/5, 3/5], [1/5, 2/5, 3/5, 4/5]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{4}{5}\) |
|---|---|
| Kissing number | 10 |
| Order of the isometry group | \(240 = 2^{4} \cdot 3 \cdot 5\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 20 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{4}\) |
| Rank of \(R(L)\) | 4 |
| \([L : R(L)]\) | 5 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/5\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{4}\) | \(\{2\}\) | 20 | \(\cong A_{4}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{4}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{3} - 2e_{4}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |