Tag 0085
\(E_{7}^{*}\)
E7*
Invariants
| Rank | 7 |
|---|---|
| Signature \((n_+, n_-)\) | (7, 0) |
| Determinant | \(\tfrac{1}{2}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{7}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | |
|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 2 | 3 | 4 | 3 | 2 | 1 |
| \(e_{2}\) | 2 | 7/2 | 4 | 6 | 9/2 | 3 | 3/2 |
| \(e_{3}\) | 3 | 4 | 6 | 8 | 6 | 4 | 2 |
| \(e_{4}\) | 4 | 6 | 8 | 12 | 9 | 6 | 3 |
| \(e_{5}\) | 3 | 9/2 | 6 | 9 | 15/2 | 5 | 5/2 |
| \(e_{6}\) | 2 | 3 | 4 | 6 | 5 | 4 | 2 |
| \(e_{7}\) | 1 | 3/2 | 2 | 3 | 5/2 | 2 | 3/2 |
[[2, 2, 3, 4, 3, 2, 1], [2, 7/2, 4, 6, 9/2, 3, 3/2], [3, 4, 6, 8, 6, 4, 2], [4, 6, 8, 12, 9, 6, 3], [3, 9/2, 6, 9, 15/2, 5, 5/2], [2, 3, 4, 6, 5, 4, 2], [1, 3/2, 2, 3, 5/2, 2, 3/2]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{3}{2}\) |
|---|---|
| Kissing number | 56 |
| Order of the isometry group | \(2903040 = 2^{10} \cdot 3^{4} \cdot 5 \cdot 7\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 126 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong E_{7}\) |
| Rank of \(R(L)\) | 7 |
| \([L : R(L)]\) | 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 | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(E_{7}\) | \(\{2\}\) | 126 | \(\cong E_{7}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} - e_{2} - e_{3} + e_{5}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{3} - e_{4} + e_{6}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{5} - 2e_{6} + e_{7}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{6} - 2e_{7}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{2} - e_{3} + e_{7}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{3} - e_{4}\) | \(2\) |