Tag 02AU

\(\Lambda_{\mathfrak{G}_{76}}(-1)\)

Lambda_G(-1), G = G_76 of Hashimoto·coinvariant lattice of the symplectic K3 group G_76 of Hashimoto, twisted by -1

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(384\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{19}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)4-2-2-211-1-2-11201-1-1122-1
\(e_{2}\)-2400-2120-1-2-2-111-11-10-1
\(e_{3}\)-2042-10112-10-1-1-11-2-2-12
\(e_{4}\)-2024-10120-101-2020-1-20
\(e_{5}\)1-2-1-14-2-1-11210-1-1101-10
\(e_{6}\)1100-240-1-2-2102-10112-1
\(e_{7}\)-1211-10410-20-2-11-11-2-1-1
\(e_{8}\)-2012-1-1141-1-11-1200-1-11
\(e_{9}\)-1-1201-20140-10-100-2-1-12
\(e_{10}\)1-2-1-12-2-2-104000-11-1100
\(e_{11}\)2-200110-1-104-1-1-100000
\(e_{12}\)0-1-1100-2100-140110200
\(e_{13}\)11-1-2-12-1-1-10-1040-10120
\(e_{14}\)-11-10-1-1120-1-1104-20-1-10
\(e_{15}\)-1-11210-100101-1-2401-10
\(e_{16}\)11-200110-2-100000411-2
\(e_{17}\)2-1-2-111-2-1-11021-11142-1
\(e_{18}\)20-1-2-12-1-1-10002-1-11240
\(e_{19}\)-1-1200-1-112000000-2-104
 
[[4, -2, -2, -2, 1, 1, -1, -2, -1, 1, 2, 0, 1, -1, -1, 1, 2, 2, -1], [-2, 4, 0, 0, -2, 1, 2, 0, -1, -2, -2, -1, 1, 1, -1, 1, -1, 0, -1], [-2, 0, 4, 2, -1, 0, 1, 1, 2, -1, 0, -1, -1, -1, 1, -2, -2, -1, 2], [-2, 0, 2, 4, -1, 0, 1, 2, 0, -1, 0, 1, -2, 0, 2, 0, -1, -2, 0], [1, -2, -1, -1, 4, -2, -1, -1, 1, 2, 1, 0, -1, -1, 1, 0, 1, -1, 0], [1, 1, 0, 0, -2, 4, 0, -1, -2, -2, 1, 0, 2, -1, 0, 1, 1, 2, -1], [-1, 2, 1, 1, -1, 0, 4, 1, 0, -2, 0, -2, -1, 1, -1, 1, -2, -1, -1], [-2, 0, 1, 2, -1, -1, 1, 4, 1, -1, -1, 1, -1, 2, 0, 0, -1, -1, 1], [-1, -1, 2, 0, 1, -2, 0, 1, 4, 0, -1, 0, -1, 0, 0, -2, -1, -1, 2], [1, -2, -1, -1, 2, -2, -2, -1, 0, 4, 0, 0, 0, -1, 1, -1, 1, 0, 0], [2, -2, 0, 0, 1, 1, 0, -1, -1, 0, 4, -1, -1, -1, 0, 0, 0, 0, 0], [0, -1, -1, 1, 0, 0, -2, 1, 0, 0, -1, 4, 0, 1, 1, 0, 2, 0, 0], [1, 1, -1, -2, -1, 2, -1, -1, -1, 0, -1, 0, 4, 0, -1, 0, 1, 2, 0], [-1, 1, -1, 0, -1, -1, 1, 2, 0, -1, -1, 1, 0, 4, -2, 0, -1, -1, 0], [-1, -1, 1, 2, 1, 0, -1, 0, 0, 1, 0, 1, -1, -2, 4, 0, 1, -1, 0], [1, 1, -2, 0, 0, 1, 1, 0, -2, -1, 0, 0, 0, 0, 0, 4, 1, 1, -2], [2, -1, -2, -1, 1, 1, -2, -1, -1, 1, 0, 2, 1, -1, 1, 1, 4, 2, -1], [2, 0, -1, -2, -1, 2, -1, -1, -1, 0, 0, 0, 2, -1, -1, 1, 2, 4, 0], [-1, -1, 2, 0, 0, -1, -1, 1, 2, 0, 0, 0, 0, 0, 0, -2, -1, 0, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/4)^{2} \oplus \mathbb{Z}/24\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)22
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 18)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 18)\, L^\Sigma(s - 9, \chi_{-384n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number6222

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 6222q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots4
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 4 \rangle^{2}\)
Rank of \(R(L)\)2
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)
\(\Phi_{2}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)

Simple roots

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{1} + e_{3} - e_{6} - e_{9} - e_{11}\)\(4\)
\(\alpha^{(2)}_{1}\)\(e_{1} + e_{2} + e_{4}\)\(4\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. The coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) is negative definite of rank 19 and has no vector of norm \(-2\); this record is \(\Lambda_G(-1)\) for the groups \(\mathfrak{G}_{76} \cong H_{192}\), \(\mathfrak{G}_{66} \cong 2^4 C_6\) of Table 10.2 of Hashimoto, whose invariant lattice \(\Lambda^G\) is in row 76 of Table 10.3.

The Gram tensor is the LLL reduction of the coinvariant lattice lattices[5,10] of Höhn and Mason: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 5 that is number 10 of rank 5 in their Table 1. It satisfies their K3 criterion \(\operatorname{rk} B \geq 5\), \(\operatorname{rk} B - \operatorname{rk} A_B \geq 2\), and among the coinvariant lattices that satisfy it, it is the only one whose rank and discriminant form are those of \(\Lambda_G(-1)\): its discriminant form is the form \(q\) of row 76 of Table 10.3. The sum of its twist by \(-1\) and \(\Lambda^G\) therefore glues to an even unimodular lattice of signature \((3, 19)\), on which the stabilizer of \(B\) in \(\mathrm{Co}_0\) acts with this coinvariant lattice.

  • 02AJ \(\langle 1 \rangle \oplus \langle 2 \rangle \oplus \langle 3 \rangle\) A lattice \(M\) with \(M(4) \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{76}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(-1)\) is the coinvariant lattice \(\Lambda_G\) of the group \(\mathfrak{G}_{76}\) of Table 10.2 of Hashimoto.
  • 028S \(\Lambda_{24}\) The Leech lattice \(\Lambda\), in which \(L\) is the orthogonal complement of the sublattice fixed by a subgroup of \(\mathrm{Co}_0\).

Morphisms

References

  1. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.
  2. G. Höhn and G. Mason, "The 290 fixed-point sublattices of the Leech lattice", arXiv:1505.06420, Table 1 and its ancillary Magma file lattices.txt, entry lattices[5,10].