Tag 0198
\(D_{15}^{*}\)
D15*
Invariants
| Rank | 15 |
|---|---|
| Signature \((n_+, n_-)\) | (15, 0) |
| Determinant | \(\tfrac{1}{4}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{15}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | \(e_{13}\) | \(e_{14}\) | \(e_{15}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1/2 | 1/2 |
| \(e_{2}\) | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 |
| \(e_{3}\) | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3/2 | 3/2 |
| \(e_{4}\) | 1 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 |
| \(e_{5}\) | 1 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5/2 | 5/2 |
| \(e_{6}\) | 1 | 2 | 3 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 3 | 3 |
| \(e_{7}\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7/2 | 7/2 |
| \(e_{8}\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 4 | 4 |
| \(e_{9}\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 9 | 9 | 9 | 9/2 | 9/2 |
| \(e_{10}\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 10 | 10 | 10 | 5 | 5 |
| \(e_{11}\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 11 | 11 | 11/2 | 11/2 |
| \(e_{12}\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 12 | 6 | 6 |
| \(e_{13}\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 13/2 | 13/2 |
| \(e_{14}\) | 1/2 | 1 | 3/2 | 2 | 5/2 | 3 | 7/2 | 4 | 9/2 | 5 | 11/2 | 6 | 13/2 | 15/4 | 13/4 |
| \(e_{15}\) | 1/2 | 1 | 3/2 | 2 | 5/2 | 3 | 7/2 | 4 | 9/2 | 5 | 11/2 | 6 | 13/2 | 13/4 | 15/4 |
[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1/2, 1/2], [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1], [1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3/2, 3/2], [1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2], [1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5/2, 5/2], [1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 3, 3], [1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7/2, 7/2], [1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 4, 4], [1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9/2, 9/2], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 10, 10, 5, 5], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 11, 11/2, 11/2], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 6, 6], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13/2, 13/2], [1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5, 11/2, 6, 13/2, 15/4, 13/4], [1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5, 11/2, 6, 13/2, 13/4, 15/4]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(1\) |
|---|---|
| Kissing number | 30 |
| Order of the isometry group | \(42849873690624000 = 2^{26} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 450 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \mathrm{I}_{15,0}\) |
| Rank of \(R(L)\) | 15 |
| \([L : R(L)]\) | 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 | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(B_{15}\) | \(\{1, 2\}\) | 450 | \(\cong \mathrm{I}_{15,0}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{2}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{5} - 2e_{6} + e_{7}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{6} - 2e_{7} + e_{8}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{7} - 2e_{8} + e_{9}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{8} - 2e_{9} + e_{10}\) | \(2\) |
| \(\alpha_{10}\) | \(e_{9} - 2e_{10} + e_{11}\) | \(2\) |
| \(\alpha_{11}\) | \(e_{10} - 2e_{11} + e_{12}\) | \(2\) |
| \(\alpha_{12}\) | \(e_{11} - 2e_{12} + e_{13}\) | \(2\) |
| \(\alpha_{13}\) | \(e_{12} - 2e_{13} + e_{14} + e_{15}\) | \(2\) |
| \(\alpha_{14}\) | \(e_{13} - 2e_{14}\) | \(2\) |
| \(\alpha_{15}\) | \(e_{14} - e_{15}\) | \(1\) |