Tag 0157

\(A_{12}^{*}\)

A12*

Invariants

Rank12
Signature \((n_+, n_-)\)(12, 0)
Determinant\(\tfrac{1}{13}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)
\(e_{1}\)12/1311/1310/139/138/137/136/135/134/133/132/131/13
\(e_{2}\)11/1322/1320/1318/1316/1314/1312/1310/138/136/134/132/13
\(e_{3}\)10/1320/1330/1327/1324/1321/1318/1315/1312/139/136/133/13
\(e_{4}\)9/1318/1327/1336/1332/1328/1324/1320/1316/1312/138/134/13
\(e_{5}\)8/1316/1324/1332/1340/1335/1330/1325/1320/1315/1310/135/13
\(e_{6}\)7/1314/1321/1328/1335/1342/1336/1330/1324/1318/1312/136/13
\(e_{7}\)6/1312/1318/1324/1330/1336/1342/1335/1328/1321/1314/137/13
\(e_{8}\)5/1310/1315/1320/1325/1330/1335/1340/1332/1324/1316/138/13
\(e_{9}\)4/138/1312/1316/1320/1324/1328/1332/1336/1327/1318/139/13
\(e_{10}\)3/136/139/1312/1315/1318/1321/1324/1327/1330/1320/1310/13
\(e_{11}\)2/134/136/138/1310/1312/1314/1316/1318/1320/1322/1311/13
\(e_{12}\)1/132/133/134/135/136/137/138/139/1310/1311/1312/13
 
[[12/13, 11/13, 10/13, 9/13, 8/13, 7/13, 6/13, 5/13, 4/13, 3/13, 2/13, 1/13], [11/13, 22/13, 20/13, 18/13, 16/13, 14/13, 12/13, 10/13, 8/13, 6/13, 4/13, 2/13], [10/13, 20/13, 30/13, 27/13, 24/13, 21/13, 18/13, 15/13, 12/13, 9/13, 6/13, 3/13], [9/13, 18/13, 27/13, 36/13, 32/13, 28/13, 24/13, 20/13, 16/13, 12/13, 8/13, 4/13], [8/13, 16/13, 24/13, 32/13, 40/13, 35/13, 30/13, 25/13, 20/13, 15/13, 10/13, 5/13], [7/13, 14/13, 21/13, 28/13, 35/13, 42/13, 36/13, 30/13, 24/13, 18/13, 12/13, 6/13], [6/13, 12/13, 18/13, 24/13, 30/13, 36/13, 42/13, 35/13, 28/13, 21/13, 14/13, 7/13], [5/13, 10/13, 15/13, 20/13, 25/13, 30/13, 35/13, 40/13, 32/13, 24/13, 16/13, 8/13], [4/13, 8/13, 12/13, 16/13, 20/13, 24/13, 28/13, 32/13, 36/13, 27/13, 18/13, 9/13], [3/13, 6/13, 9/13, 12/13, 15/13, 18/13, 21/13, 24/13, 27/13, 30/13, 20/13, 10/13], [2/13, 4/13, 6/13, 8/13, 10/13, 12/13, 14/13, 16/13, 18/13, 20/13, 22/13, 11/13], [1/13, 2/13, 3/13, 4/13, 5/13, 6/13, 7/13, 8/13, 9/13, 10/13, 11/13, 12/13]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{12}{13}\)
Kissing number26
Order of the isometry group\(12454041600 = 2^{11} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots156
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{12}\)
Rank of \(R(L)\)12
\([L : R(L)]\)13
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/13\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{12}\)\(\{2\}\)156\(\cong A_{12}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{12}\)\(2\)
\(\alpha_{2}\)\(e_{11} - 2e_{12}\)\(2\)
\(\alpha_{3}\)\(e_{10} - 2e_{11} + e_{12}\)\(2\)
\(\alpha_{4}\)\(e_{9} - 2e_{10} + e_{11}\)\(2\)
\(\alpha_{5}\)\(e_{8} - 2e_{9} + e_{10}\)\(2\)
\(\alpha_{6}\)\(e_{7} - 2e_{8} + e_{9}\)\(2\)
\(\alpha_{7}\)\(e_{6} - 2e_{7} + e_{8}\)\(2\)
\(\alpha_{8}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{9}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{10}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{11}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{12}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0147 \(A_{12}\) The dual lattice \(L^*\).