Tag 0070
\(E_{6}^{*}\)
E6*
Invariants
| Rank | 6 |
|---|---|
| Signature \((n_+, n_-)\) | (6, 0) |
| Determinant | \(\tfrac{1}{3}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{6}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | |
|---|---|---|---|---|---|---|
| \(e_{1}\) | 4/3 | 1 | 5/3 | 2 | 4/3 | 2/3 |
| \(e_{2}\) | 1 | 2 | 2 | 3 | 2 | 1 |
| \(e_{3}\) | 5/3 | 2 | 10/3 | 4 | 8/3 | 4/3 |
| \(e_{4}\) | 2 | 3 | 4 | 6 | 4 | 2 |
| \(e_{5}\) | 4/3 | 2 | 8/3 | 4 | 10/3 | 5/3 |
| \(e_{6}\) | 2/3 | 1 | 4/3 | 2 | 5/3 | 4/3 |
[[4/3, 1, 5/3, 2, 4/3, 2/3], [1, 2, 2, 3, 2, 1], [5/3, 2, 10/3, 4, 8/3, 4/3], [2, 3, 4, 6, 4, 2], [4/3, 2, 8/3, 4, 10/3, 5/3], [2/3, 1, 4/3, 2, 5/3, 4/3]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{4}{3}\) |
|---|---|
| Kissing number | 54 |
| Order of the isometry group | \(103680 = 2^{8} \cdot 3^{4} \cdot 5\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 72 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong E_{6}\) |
| Rank of \(R(L)\) | 6 |
| \([L : R(L)]\) | 3 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/3\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(E_{6}\) | \(\{2\}\) | 72 | \(\cong E_{6}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{2} - e_{4}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{1} - e_{2} - e_{3} + e_{5}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{3} - e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{5} - 2e_{6}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{2} - e_{3} + e_{6}\) | \(2\) |