Tag 0027
\(A_{3}^{*}\)
A3*
Invariants
| Rank | 3 |
|---|---|
| Signature \((n_+, n_-)\) | (3, 0) |
| Determinant | \(\tfrac{1}{4}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}, e_{3}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | |
|---|---|---|---|
| \(e_{1}\) | 3/4 | 1/2 | 1/4 |
| \(e_{2}\) | 1/2 | 1 | 1/2 |
| \(e_{3}\) | 1/4 | 1/2 | 3/4 |
[[3/4, 1/2, 1/4], [1/2, 1, 1/2], [1/4, 1/2, 3/4]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{3}{4}\) |
|---|---|
| Kissing number | 8 |
| Order of the isometry group | \(48 = 2^{4} \cdot 3\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 18 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \mathrm{I}_{3,0}\) |
| Rank of \(R(L)\) | 3 |
| \([L : R(L)]\) | 2 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/2\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(B_{3}\) | \(\{1, 2\}\) | 18 | \(\cong \mathrm{I}_{3,0}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{2} - 2e_{3}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{2}\) | \(1\) |