Tag 0129

\(A_{10}^{*}\)

A10*

Invariants

Rank10
Signature \((n_+, n_-)\)(10, 0)
Determinant\(\tfrac{1}{11}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{10}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)
\(e_{1}\)10/119/118/117/116/115/114/113/112/111/11
\(e_{2}\)9/1118/1116/1114/1112/1110/118/116/114/112/11
\(e_{3}\)8/1116/1124/1121/1118/1115/1112/119/116/113/11
\(e_{4}\)7/1114/1121/1128/1124/1120/1116/1112/118/114/11
\(e_{5}\)6/1112/1118/1124/1130/1125/1120/1115/1110/115/11
\(e_{6}\)5/1110/1115/1120/1125/1130/1124/1118/1112/116/11
\(e_{7}\)4/118/1112/1116/1120/1124/1128/1121/1114/117/11
\(e_{8}\)3/116/119/1112/1115/1118/1121/1124/1116/118/11
\(e_{9}\)2/114/116/118/1110/1112/1114/1116/1118/119/11
\(e_{10}\)1/112/113/114/115/116/117/118/119/1110/11
 
[[10/11, 9/11, 8/11, 7/11, 6/11, 5/11, 4/11, 3/11, 2/11, 1/11], [9/11, 18/11, 16/11, 14/11, 12/11, 10/11, 8/11, 6/11, 4/11, 2/11], [8/11, 16/11, 24/11, 21/11, 18/11, 15/11, 12/11, 9/11, 6/11, 3/11], [7/11, 14/11, 21/11, 28/11, 24/11, 20/11, 16/11, 12/11, 8/11, 4/11], [6/11, 12/11, 18/11, 24/11, 30/11, 25/11, 20/11, 15/11, 10/11, 5/11], [5/11, 10/11, 15/11, 20/11, 25/11, 30/11, 24/11, 18/11, 12/11, 6/11], [4/11, 8/11, 12/11, 16/11, 20/11, 24/11, 28/11, 21/11, 14/11, 7/11], [3/11, 6/11, 9/11, 12/11, 15/11, 18/11, 21/11, 24/11, 16/11, 8/11], [2/11, 4/11, 6/11, 8/11, 10/11, 12/11, 14/11, 16/11, 18/11, 9/11], [1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{10}{11}\)
Kissing number22
Order of the isometry group\(79833600 = 2^{9} \cdot 3^{4} \cdot 5^{2} \cdot 7 \cdot 11\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots110
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{10}\)
Rank of \(R(L)\)10
\([L : R(L)]\)11
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/11\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{10}\)\(\{2\}\)110\(\cong A_{10}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{10}\)\(2\)
\(\alpha_{2}\)\(e_{9} - 2e_{10}\)\(2\)
\(\alpha_{3}\)\(e_{8} - 2e_{9} + e_{10}\)\(2\)
\(\alpha_{4}\)\(e_{7} - 2e_{8} + e_{9}\)\(2\)
\(\alpha_{5}\)\(e_{6} - 2e_{7} + e_{8}\)\(2\)
\(\alpha_{6}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{7}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{8}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{9}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{10}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0121 \(A_{10}\) The dual lattice \(L^*\).