Tag 0170

\(A_{13}^{*}\)

A13*

Invariants

Rank13
Signature \((n_+, n_-)\)(13, 0)
Determinant\(\tfrac{1}{14}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{13}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)
\(e_{1}\)13/146/711/145/79/144/71/23/75/142/73/141/71/14
\(e_{2}\)6/712/711/710/79/78/716/75/74/73/72/71/7
\(e_{3}\)11/1411/733/1415/727/1412/73/29/715/146/79/143/73/14
\(e_{4}\)5/710/715/720/718/716/7212/710/78/76/74/72/7
\(e_{5}\)9/149/727/1418/745/1420/75/215/725/1410/715/145/75/14
\(e_{6}\)4/78/712/716/720/724/7318/715/712/79/76/73/7
\(e_{7}\)1/213/225/237/235/223/211/2
\(e_{8}\)3/76/79/712/715/718/7324/720/716/712/78/74/7
\(e_{9}\)5/145/715/1410/725/1415/75/220/745/1418/727/149/79/14
\(e_{10}\)2/74/76/78/710/712/7216/718/720/715/710/75/7
\(e_{11}\)3/143/79/146/715/149/73/212/727/1415/733/1411/711/14
\(e_{12}\)1/72/73/74/75/76/718/79/710/711/712/76/7
\(e_{13}\)1/141/73/142/75/143/71/24/79/145/711/146/713/14
 
[[13/14, 6/7, 11/14, 5/7, 9/14, 4/7, 1/2, 3/7, 5/14, 2/7, 3/14, 1/7, 1/14], [6/7, 12/7, 11/7, 10/7, 9/7, 8/7, 1, 6/7, 5/7, 4/7, 3/7, 2/7, 1/7], [11/14, 11/7, 33/14, 15/7, 27/14, 12/7, 3/2, 9/7, 15/14, 6/7, 9/14, 3/7, 3/14], [5/7, 10/7, 15/7, 20/7, 18/7, 16/7, 2, 12/7, 10/7, 8/7, 6/7, 4/7, 2/7], [9/14, 9/7, 27/14, 18/7, 45/14, 20/7, 5/2, 15/7, 25/14, 10/7, 15/14, 5/7, 5/14], [4/7, 8/7, 12/7, 16/7, 20/7, 24/7, 3, 18/7, 15/7, 12/7, 9/7, 6/7, 3/7], [1/2, 1, 3/2, 2, 5/2, 3, 7/2, 3, 5/2, 2, 3/2, 1, 1/2], [3/7, 6/7, 9/7, 12/7, 15/7, 18/7, 3, 24/7, 20/7, 16/7, 12/7, 8/7, 4/7], [5/14, 5/7, 15/14, 10/7, 25/14, 15/7, 5/2, 20/7, 45/14, 18/7, 27/14, 9/7, 9/14], [2/7, 4/7, 6/7, 8/7, 10/7, 12/7, 2, 16/7, 18/7, 20/7, 15/7, 10/7, 5/7], [3/14, 3/7, 9/14, 6/7, 15/14, 9/7, 3/2, 12/7, 27/14, 15/7, 33/14, 11/7, 11/14], [1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7, 9/7, 10/7, 11/7, 12/7, 6/7], [1/14, 1/7, 3/14, 2/7, 5/14, 3/7, 1/2, 4/7, 9/14, 5/7, 11/14, 6/7, 13/14]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{13}{14}\)
Kissing number28
Order of the isometry group\(174356582400 = 2^{12} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots182
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{13}\)
Rank of \(R(L)\)13
\([L : R(L)]\)14
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/14\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{13}\)\(\{2\}\)182\(\cong A_{13}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{13}\)\(2\)
\(\alpha_{2}\)\(e_{12} - 2e_{13}\)\(2\)
\(\alpha_{3}\)\(e_{11} - 2e_{12} + e_{13}\)\(2\)
\(\alpha_{4}\)\(e_{10} - 2e_{11} + e_{12}\)\(2\)
\(\alpha_{5}\)\(e_{9} - 2e_{10} + e_{11}\)\(2\)
\(\alpha_{6}\)\(e_{8} - 2e_{9} + e_{10}\)\(2\)
\(\alpha_{7}\)\(e_{7} - 2e_{8} + e_{9}\)\(2\)
\(\alpha_{8}\)\(e_{6} - 2e_{7} + e_{8}\)\(2\)
\(\alpha_{9}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{10}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{11}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{12}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{13}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0164 \(A_{13}\) The dual lattice \(L^*\).