Tag 0211

\(A_{16}^{*}\)

A16*

Invariants

Rank16
Signature \((n_+, n_-)\)(16, 0)
Determinant\(\tfrac{1}{17}\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{16}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)
\(e_{1}\)16/1715/1714/1713/1712/1711/1710/179/178/177/176/175/174/173/172/171/17
\(e_{2}\)15/1730/1728/1726/1724/1722/1720/1718/1716/1714/1712/1710/178/176/174/172/17
\(e_{3}\)14/1728/1742/1739/1736/1733/1730/1727/1724/1721/1718/1715/1712/179/176/173/17
\(e_{4}\)13/1726/1739/1752/1748/1744/1740/1736/1732/1728/1724/1720/1716/1712/178/174/17
\(e_{5}\)12/1724/1736/1748/1760/1755/1750/1745/1740/1735/1730/1725/1720/1715/1710/175/17
\(e_{6}\)11/1722/1733/1744/1755/1766/1760/1754/1748/1742/1736/1730/1724/1718/1712/176/17
\(e_{7}\)10/1720/1730/1740/1750/1760/1770/1763/1756/1749/1742/1735/1728/1721/1714/177/17
\(e_{8}\)9/1718/1727/1736/1745/1754/1763/1772/1764/1756/1748/1740/1732/1724/1716/178/17
\(e_{9}\)8/1716/1724/1732/1740/1748/1756/1764/1772/1763/1754/1745/1736/1727/1718/179/17
\(e_{10}\)7/1714/1721/1728/1735/1742/1749/1756/1763/1770/1760/1750/1740/1730/1720/1710/17
\(e_{11}\)6/1712/1718/1724/1730/1736/1742/1748/1754/1760/1766/1755/1744/1733/1722/1711/17
\(e_{12}\)5/1710/1715/1720/1725/1730/1735/1740/1745/1750/1755/1760/1748/1736/1724/1712/17
\(e_{13}\)4/178/1712/1716/1720/1724/1728/1732/1736/1740/1744/1748/1752/1739/1726/1713/17
\(e_{14}\)3/176/179/1712/1715/1718/1721/1724/1727/1730/1733/1736/1739/1742/1728/1714/17
\(e_{15}\)2/174/176/178/1710/1712/1714/1716/1718/1720/1722/1724/1726/1728/1730/1715/17
\(e_{16}\)1/172/173/174/175/176/177/178/179/1710/1711/1712/1713/1714/1715/1716/17
 
[[16/17, 15/17, 14/17, 13/17, 12/17, 11/17, 10/17, 9/17, 8/17, 7/17, 6/17, 5/17, 4/17, 3/17, 2/17, 1/17], [15/17, 30/17, 28/17, 26/17, 24/17, 22/17, 20/17, 18/17, 16/17, 14/17, 12/17, 10/17, 8/17, 6/17, 4/17, 2/17], [14/17, 28/17, 42/17, 39/17, 36/17, 33/17, 30/17, 27/17, 24/17, 21/17, 18/17, 15/17, 12/17, 9/17, 6/17, 3/17], [13/17, 26/17, 39/17, 52/17, 48/17, 44/17, 40/17, 36/17, 32/17, 28/17, 24/17, 20/17, 16/17, 12/17, 8/17, 4/17], [12/17, 24/17, 36/17, 48/17, 60/17, 55/17, 50/17, 45/17, 40/17, 35/17, 30/17, 25/17, 20/17, 15/17, 10/17, 5/17], [11/17, 22/17, 33/17, 44/17, 55/17, 66/17, 60/17, 54/17, 48/17, 42/17, 36/17, 30/17, 24/17, 18/17, 12/17, 6/17], [10/17, 20/17, 30/17, 40/17, 50/17, 60/17, 70/17, 63/17, 56/17, 49/17, 42/17, 35/17, 28/17, 21/17, 14/17, 7/17], [9/17, 18/17, 27/17, 36/17, 45/17, 54/17, 63/17, 72/17, 64/17, 56/17, 48/17, 40/17, 32/17, 24/17, 16/17, 8/17], [8/17, 16/17, 24/17, 32/17, 40/17, 48/17, 56/17, 64/17, 72/17, 63/17, 54/17, 45/17, 36/17, 27/17, 18/17, 9/17], [7/17, 14/17, 21/17, 28/17, 35/17, 42/17, 49/17, 56/17, 63/17, 70/17, 60/17, 50/17, 40/17, 30/17, 20/17, 10/17], [6/17, 12/17, 18/17, 24/17, 30/17, 36/17, 42/17, 48/17, 54/17, 60/17, 66/17, 55/17, 44/17, 33/17, 22/17, 11/17], [5/17, 10/17, 15/17, 20/17, 25/17, 30/17, 35/17, 40/17, 45/17, 50/17, 55/17, 60/17, 48/17, 36/17, 24/17, 12/17], [4/17, 8/17, 12/17, 16/17, 20/17, 24/17, 28/17, 32/17, 36/17, 40/17, 44/17, 48/17, 52/17, 39/17, 26/17, 13/17], [3/17, 6/17, 9/17, 12/17, 15/17, 18/17, 21/17, 24/17, 27/17, 30/17, 33/17, 36/17, 39/17, 42/17, 28/17, 14/17], [2/17, 4/17, 6/17, 8/17, 10/17, 12/17, 14/17, 16/17, 18/17, 20/17, 22/17, 24/17, 26/17, 28/17, 30/17, 15/17], [1/17, 2/17, 3/17, 4/17, 5/17, 6/17, 7/17, 8/17, 9/17, 10/17, 11/17, 12/17, 13/17, 14/17, 15/17, 16/17]]

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(\tfrac{16}{17}\)
Kissing number34
Order of the isometry group\(711374856192000 = 2^{16} \cdot 3^{6} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17\)

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots272
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong A_{16}\)
Rank of \(R(L)\)16
\([L : R(L)]\)17
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/17\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{16}\)\(\{2\}\)272\(\cong A_{16}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1} + e_{16}\)\(2\)
\(\alpha_{2}\)\(e_{15} - 2e_{16}\)\(2\)
\(\alpha_{3}\)\(e_{14} - 2e_{15} + e_{16}\)\(2\)
\(\alpha_{4}\)\(e_{13} - 2e_{14} + e_{15}\)\(2\)
\(\alpha_{5}\)\(e_{12} - 2e_{13} + e_{14}\)\(2\)
\(\alpha_{6}\)\(e_{11} - 2e_{12} + e_{13}\)\(2\)
\(\alpha_{7}\)\(e_{10} - 2e_{11} + e_{12}\)\(2\)
\(\alpha_{8}\)\(e_{9} - 2e_{10} + e_{11}\)\(2\)
\(\alpha_{9}\)\(e_{8} - 2e_{9} + e_{10}\)\(2\)
\(\alpha_{10}\)\(e_{7} - 2e_{8} + e_{9}\)\(2\)
\(\alpha_{11}\)\(e_{6} - 2e_{7} + e_{8}\)\(2\)
\(\alpha_{12}\)\(e_{5} - 2e_{6} + e_{7}\)\(2\)
\(\alpha_{13}\)\(e_{4} - 2e_{5} + e_{6}\)\(2\)
\(\alpha_{14}\)\(e_{3} - 2e_{4} + e_{5}\)\(2\)
\(\alpha_{15}\)\(e_{2} - 2e_{3} + e_{4}\)\(2\)
\(\alpha_{16}\)\(e_{1} - 2e_{2} + e_{3}\)\(2\)
  • 0203 \(A_{16}\) The dual lattice \(L^*\).