Tag 0017
\(A_{2}^{*}\)
A2*
Invariants
| Rank | 2 |
|---|---|
| Signature \((n_+, n_-)\) | (2, 0) |
| Determinant | \(\tfrac{1}{3}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}\).
| \(e_{1}\) | \(e_{2}\) | |
|---|---|---|
| \(e_{1}\) | 2/3 | 1/3 |
| \(e_{2}\) | 1/3 | 2/3 |
[[2/3, 1/3], [1/3, 2/3]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{2}{3}\) |
|---|---|
| Kissing number | 6 |
| Order of the isometry group | \(12 = 2^{2} \cdot 3\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{\tfrac{2}{3}\}\) |
| Number of roots | 12 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{2}(\tfrac{1}{3})\) |
| Rank of \(R(L)\) | 2 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(G_{2}\) | \(\{\tfrac{2}{3}, 2\}\) | 12 | \(\cong A_{2}(\tfrac{1}{3})\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{2}\) | \(\tfrac{2}{3}\) |
| \(\alpha_{2}\) | \(e_{1} - 2e_{2}\) | \(2\) |