Tag 0141
\(A_{11}^{*}\)
A11*
Invariants
| Rank | 11 |
|---|---|
| Signature \((n_+, n_-)\) | (11, 0) |
| Determinant | \(\tfrac{1}{12}\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{11}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 11/12 | 5/6 | 3/4 | 2/3 | 7/12 | 1/2 | 5/12 | 1/3 | 1/4 | 1/6 | 1/12 |
| \(e_{2}\) | 5/6 | 5/3 | 3/2 | 4/3 | 7/6 | 1 | 5/6 | 2/3 | 1/2 | 1/3 | 1/6 |
| \(e_{3}\) | 3/4 | 3/2 | 9/4 | 2 | 7/4 | 3/2 | 5/4 | 1 | 3/4 | 1/2 | 1/4 |
| \(e_{4}\) | 2/3 | 4/3 | 2 | 8/3 | 7/3 | 2 | 5/3 | 4/3 | 1 | 2/3 | 1/3 |
| \(e_{5}\) | 7/12 | 7/6 | 7/4 | 7/3 | 35/12 | 5/2 | 25/12 | 5/3 | 5/4 | 5/6 | 5/12 |
| \(e_{6}\) | 1/2 | 1 | 3/2 | 2 | 5/2 | 3 | 5/2 | 2 | 3/2 | 1 | 1/2 |
| \(e_{7}\) | 5/12 | 5/6 | 5/4 | 5/3 | 25/12 | 5/2 | 35/12 | 7/3 | 7/4 | 7/6 | 7/12 |
| \(e_{8}\) | 1/3 | 2/3 | 1 | 4/3 | 5/3 | 2 | 7/3 | 8/3 | 2 | 4/3 | 2/3 |
| \(e_{9}\) | 1/4 | 1/2 | 3/4 | 1 | 5/4 | 3/2 | 7/4 | 2 | 9/4 | 3/2 | 3/4 |
| \(e_{10}\) | 1/6 | 1/3 | 1/2 | 2/3 | 5/6 | 1 | 7/6 | 4/3 | 3/2 | 5/3 | 5/6 |
| \(e_{11}\) | 1/12 | 1/6 | 1/4 | 1/3 | 5/12 | 1/2 | 7/12 | 2/3 | 3/4 | 5/6 | 11/12 |
[[11/12, 5/6, 3/4, 2/3, 7/12, 1/2, 5/12, 1/3, 1/4, 1/6, 1/12], [5/6, 5/3, 3/2, 4/3, 7/6, 1, 5/6, 2/3, 1/2, 1/3, 1/6], [3/4, 3/2, 9/4, 2, 7/4, 3/2, 5/4, 1, 3/4, 1/2, 1/4], [2/3, 4/3, 2, 8/3, 7/3, 2, 5/3, 4/3, 1, 2/3, 1/3], [7/12, 7/6, 7/4, 7/3, 35/12, 5/2, 25/12, 5/3, 5/4, 5/6, 5/12], [1/2, 1, 3/2, 2, 5/2, 3, 5/2, 2, 3/2, 1, 1/2], [5/12, 5/6, 5/4, 5/3, 25/12, 5/2, 35/12, 7/3, 7/4, 7/6, 7/12], [1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 2, 4/3, 2/3], [1/4, 1/2, 3/4, 1, 5/4, 3/2, 7/4, 2, 9/4, 3/2, 3/4], [1/6, 1/3, 1/2, 2/3, 5/6, 1, 7/6, 4/3, 3/2, 5/3, 5/6], [1/12, 1/6, 1/4, 1/3, 5/12, 1/2, 7/12, 2/3, 3/4, 5/6, 11/12]]
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(\tfrac{11}{12}\) |
|---|---|
| Kissing number | 24 |
| Order of the isometry group | \(958003200 = 2^{11} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11\) |
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 132 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{11}\) |
| Rank of \(R(L)\) | 11 |
| \([L : R(L)]\) | 12 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/12\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{11}\) | \(\{2\}\) | 132 | \(\cong A_{11}\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1} + e_{11}\) | \(2\) |
| \(\alpha_{2}\) | \(e_{10} - 2e_{11}\) | \(2\) |
| \(\alpha_{3}\) | \(e_{9} - 2e_{10} + e_{11}\) | \(2\) |
| \(\alpha_{4}\) | \(e_{8} - 2e_{9} + e_{10}\) | \(2\) |
| \(\alpha_{5}\) | \(e_{7} - 2e_{8} + e_{9}\) | \(2\) |
| \(\alpha_{6}\) | \(e_{6} - 2e_{7} + e_{8}\) | \(2\) |
| \(\alpha_{7}\) | \(e_{5} - 2e_{6} + e_{7}\) | \(2\) |
| \(\alpha_{8}\) | \(e_{4} - 2e_{5} + e_{6}\) | \(2\) |
| \(\alpha_{9}\) | \(e_{3} - 2e_{4} + e_{5}\) | \(2\) |
| \(\alpha_{10}\) | \(e_{2} - 2e_{3} + e_{4}\) | \(2\) |
| \(\alpha_{11}\) | \(e_{1} - 2e_{2} + e_{3}\) | \(2\) |