Tag 0083

\(A_{7}^{*}\)

A7*

Invariants

Rank7
Signature \((n_+, n_-)\)(7, 0)
Determinant\(\tfrac{1}{8}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{7}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)
\(e_{1}\)7/83/45/81/23/81/41/8
\(e_{2}\)3/43/25/413/41/21/4
\(e_{3}\)5/85/415/83/29/83/43/8
\(e_{4}\)1/213/223/211/2
\(e_{5}\)3/83/49/83/215/85/45/8
\(e_{6}\)1/41/23/415/43/23/4
\(e_{7}\)1/81/43/81/25/83/47/8
 
[[7/8, 3/4, 5/8, 1/2, 3/8, 1/4, 1/8], [3/4, 3/2, 5/4, 1, 3/4, 1/2, 1/4], [5/8, 5/4, 15/8, 3/2, 9/8, 3/4, 3/8], [1/2, 1, 3/2, 2, 3/2, 1, 1/2], [3/8, 3/4, 9/8, 3/2, 15/8, 5/4, 5/8], [1/4, 1/2, 3/4, 1, 5/4, 3/2, 3/4], [1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{7}{8}\)
Kissing number16
Order of the isometry group\(80640 = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots56
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{7}\)
Rank of \(R(L)\)7
\([L : R(L)]\)8
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/8\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{7}\)\(\{2\}\)56\(\cong A_{7}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{7}\)\(2\)
\(\alpha_{2}\)\(e_{6} - 2e_{7}\)\(2\)
\(\alpha_{3}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{4}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{5}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{6}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{7}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0077 \(A_{7}\) The dual lattice \(L^*\).