Tag 0197

\(A_{15}^{*}\)

A15*

Invariants

Rank15
Signature \((n_+, n_-)\)(15, 0)
Determinant\(\tfrac{1}{16}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{15}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)
\(e_{1}\)15/167/813/163/411/165/89/161/27/163/85/161/43/161/81/16
\(e_{2}\)7/87/413/83/211/85/49/817/83/45/81/23/81/41/8
\(e_{3}\)13/1613/839/169/433/1615/827/163/221/169/815/163/49/163/83/16
\(e_{4}\)3/43/29/4311/45/29/427/43/25/413/41/21/4
\(e_{5}\)11/1611/833/1611/455/1625/845/165/235/1615/825/165/415/165/85/16
\(e_{6}\)5/85/415/85/225/815/427/8321/89/415/83/29/83/43/8
\(e_{7}\)9/169/827/169/445/1627/863/167/249/1621/835/167/421/167/87/16
\(e_{8}\)1/213/225/237/247/235/223/211/2
\(e_{9}\)7/167/821/167/435/1621/849/167/263/1627/845/169/427/169/89/16
\(e_{10}\)3/83/49/83/215/89/421/8327/815/425/85/215/85/45/8
\(e_{11}\)5/165/815/165/425/1615/835/165/245/1625/855/1611/433/1611/811/16
\(e_{12}\)1/41/23/415/43/27/429/45/211/439/43/23/4
\(e_{13}\)3/163/89/163/415/169/821/163/227/1615/833/169/439/1613/813/16
\(e_{14}\)1/81/43/81/25/83/47/819/85/411/83/213/87/47/8
\(e_{15}\)1/161/83/161/45/163/87/161/29/165/811/163/413/167/815/16
 
[[15/16, 7/8, 13/16, 3/4, 11/16, 5/8, 9/16, 1/2, 7/16, 3/8, 5/16, 1/4, 3/16, 1/8, 1/16], [7/8, 7/4, 13/8, 3/2, 11/8, 5/4, 9/8, 1, 7/8, 3/4, 5/8, 1/2, 3/8, 1/4, 1/8], [13/16, 13/8, 39/16, 9/4, 33/16, 15/8, 27/16, 3/2, 21/16, 9/8, 15/16, 3/4, 9/16, 3/8, 3/16], [3/4, 3/2, 9/4, 3, 11/4, 5/2, 9/4, 2, 7/4, 3/2, 5/4, 1, 3/4, 1/2, 1/4], [11/16, 11/8, 33/16, 11/4, 55/16, 25/8, 45/16, 5/2, 35/16, 15/8, 25/16, 5/4, 15/16, 5/8, 5/16], [5/8, 5/4, 15/8, 5/2, 25/8, 15/4, 27/8, 3, 21/8, 9/4, 15/8, 3/2, 9/8, 3/4, 3/8], [9/16, 9/8, 27/16, 9/4, 45/16, 27/8, 63/16, 7/2, 49/16, 21/8, 35/16, 7/4, 21/16, 7/8, 7/16], [1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 7/2, 3, 5/2, 2, 3/2, 1, 1/2], [7/16, 7/8, 21/16, 7/4, 35/16, 21/8, 49/16, 7/2, 63/16, 27/8, 45/16, 9/4, 27/16, 9/8, 9/16], [3/8, 3/4, 9/8, 3/2, 15/8, 9/4, 21/8, 3, 27/8, 15/4, 25/8, 5/2, 15/8, 5/4, 5/8], [5/16, 5/8, 15/16, 5/4, 25/16, 15/8, 35/16, 5/2, 45/16, 25/8, 55/16, 11/4, 33/16, 11/8, 11/16], [1/4, 1/2, 3/4, 1, 5/4, 3/2, 7/4, 2, 9/4, 5/2, 11/4, 3, 9/4, 3/2, 3/4], [3/16, 3/8, 9/16, 3/4, 15/16, 9/8, 21/16, 3/2, 27/16, 15/8, 33/16, 9/4, 39/16, 13/8, 13/16], [1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, 1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 7/8], [1/16, 1/8, 3/16, 1/4, 5/16, 3/8, 7/16, 1/2, 9/16, 5/8, 11/16, 3/4, 13/16, 7/8, 15/16]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{15}{16}\)
Kissing number32
Order of the isometry group\(41845579776000 = 2^{16} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots240
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{15}\)
Rank of \(R(L)\)15
\([L : R(L)]\)16
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/16\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{15}\)\(\{2\}\)240\(\cong A_{15}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{15}\)\(2\)
\(\alpha_{2}\)\(e_{14} - 2e_{15}\)\(2\)
\(\alpha_{3}\)\(e_{13} - 2e_{14} + e_{15}\)\(2\)
\(\alpha_{4}\)\(e_{12} - 2e_{13} + e_{14}\)\(2\)
\(\alpha_{5}\)\(e_{11} - 2e_{12} + e_{13}\)\(2\)
\(\alpha_{6}\)\(e_{10} - 2e_{11} + e_{12}\)\(2\)
\(\alpha_{7}\)\(e_{9} - 2e_{10} + e_{11}\)\(2\)
\(\alpha_{8}\)\(e_{8} - 2e_{9} + e_{10}\)\(2\)
\(\alpha_{9}\)\(e_{7} - 2e_{8} + e_{9}\)\(2\)
\(\alpha_{10}\)\(e_{6} - 2e_{7} + e_{8}\)\(2\)
\(\alpha_{11}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{12}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{13}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{14}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{15}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0191 \(A_{15}\) The dual lattice \(L^*\).