Tag 0184

\(A_{14}^{*}\)

A14*

Invariants

Rank14
Signature \((n_+, n_-)\)(14, 0)
Determinant\(\tfrac{1}{15}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{14}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)
\(e_{1}\)14/1513/154/511/152/33/58/157/152/51/34/151/52/151/15
\(e_{2}\)13/1526/158/522/154/36/516/1514/154/52/38/152/54/152/15
\(e_{3}\)4/58/512/511/529/58/57/56/514/53/52/51/5
\(e_{4}\)11/1522/1511/544/158/312/532/1528/158/54/316/154/58/154/15
\(e_{5}\)2/34/328/310/338/37/325/34/312/31/3
\(e_{6}\)3/56/59/512/5318/516/514/512/528/56/54/52/5
\(e_{7}\)8/1516/158/532/158/316/556/1549/1514/57/328/157/514/157/15
\(e_{8}\)7/1514/157/528/157/314/549/1556/1516/58/332/158/516/158/15
\(e_{9}\)2/54/56/58/5212/514/516/518/5312/59/56/53/5
\(e_{10}\)1/32/314/35/327/38/3310/38/324/32/3
\(e_{11}\)4/158/154/516/154/38/528/1532/1512/58/344/1511/522/1511/15
\(e_{12}\)1/52/53/54/516/57/58/59/5211/512/58/54/5
\(e_{13}\)2/154/152/58/152/34/514/1516/156/54/322/158/526/1513/15
\(e_{14}\)1/152/151/54/151/32/57/158/153/52/311/154/513/1514/15
 
[[14/15, 13/15, 4/5, 11/15, 2/3, 3/5, 8/15, 7/15, 2/5, 1/3, 4/15, 1/5, 2/15, 1/15], [13/15, 26/15, 8/5, 22/15, 4/3, 6/5, 16/15, 14/15, 4/5, 2/3, 8/15, 2/5, 4/15, 2/15], [4/5, 8/5, 12/5, 11/5, 2, 9/5, 8/5, 7/5, 6/5, 1, 4/5, 3/5, 2/5, 1/5], [11/15, 22/15, 11/5, 44/15, 8/3, 12/5, 32/15, 28/15, 8/5, 4/3, 16/15, 4/5, 8/15, 4/15], [2/3, 4/3, 2, 8/3, 10/3, 3, 8/3, 7/3, 2, 5/3, 4/3, 1, 2/3, 1/3], [3/5, 6/5, 9/5, 12/5, 3, 18/5, 16/5, 14/5, 12/5, 2, 8/5, 6/5, 4/5, 2/5], [8/15, 16/15, 8/5, 32/15, 8/3, 16/5, 56/15, 49/15, 14/5, 7/3, 28/15, 7/5, 14/15, 7/15], [7/15, 14/15, 7/5, 28/15, 7/3, 14/5, 49/15, 56/15, 16/5, 8/3, 32/15, 8/5, 16/15, 8/15], [2/5, 4/5, 6/5, 8/5, 2, 12/5, 14/5, 16/5, 18/5, 3, 12/5, 9/5, 6/5, 3/5], [1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 8/3, 2, 4/3, 2/3], [4/15, 8/15, 4/5, 16/15, 4/3, 8/5, 28/15, 32/15, 12/5, 8/3, 44/15, 11/5, 22/15, 11/15], [1/5, 2/5, 3/5, 4/5, 1, 6/5, 7/5, 8/5, 9/5, 2, 11/5, 12/5, 8/5, 4/5], [2/15, 4/15, 2/5, 8/15, 2/3, 4/5, 14/15, 16/15, 6/5, 4/3, 22/15, 8/5, 26/15, 13/15], [1/15, 2/15, 1/5, 4/15, 1/3, 2/5, 7/15, 8/15, 3/5, 2/3, 11/15, 4/5, 13/15, 14/15]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{14}{15}\)
Kissing number30
Order of the isometry group\(2615348736000 = 2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots210
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{14}\)
Rank of \(R(L)\)14
\([L : R(L)]\)15
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/15\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{14}\)\(\{2\}\)210\(\cong A_{14}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{14}\)\(2\)
\(\alpha_{2}\)\(e_{13} - 2e_{14}\)\(2\)
\(\alpha_{3}\)\(e_{12} - 2e_{13} + e_{14}\)\(2\)
\(\alpha_{4}\)\(e_{11} - 2e_{12} + e_{13}\)\(2\)
\(\alpha_{5}\)\(e_{10} - 2e_{11} + e_{12}\)\(2\)
\(\alpha_{6}\)\(e_{9} - 2e_{10} + e_{11}\)\(2\)
\(\alpha_{7}\)\(e_{8} - 2e_{9} + e_{10}\)\(2\)
\(\alpha_{8}\)\(e_{7} - 2e_{8} + e_{9}\)\(2\)
\(\alpha_{9}\)\(e_{6} - 2e_{7} + e_{8}\)\(2\)
\(\alpha_{10}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{11}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{12}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{13}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{14}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0176 \(A_{14}\) The dual lattice \(L^*\).