Tag 02BM

\(\Lambda_{\mathfrak{G}_{10}}(-1)\)

Lambda_G(-1), G = G_10 of Hashimoto·coinvariant lattice of the symplectic K3 group G_10 of Hashimoto, twisted by -1

Invariants

Rank15
Signature \((n_+, n_-)\)(15, 0)
Determinant\(1024\)

Gram tensor

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)
\(e_{1}\)4-2-22-21-1210-2202-2
\(e_{2}\)-242-22-11-21-12-1102
\(e_{3}\)-22400-220012-1-1-11
\(e_{4}\)2-204-2-1121000-21-1
\(e_{5}\)-220-2411-2-1-11-1002
\(e_{6}\)1-1-2-114-2-1-2-1-2000-1
\(e_{7}\)-11211-2410120-201
\(e_{8}\)2-202-2-1141101-11-1
\(e_{9}\)1101-1-2014-110120
\(e_{10}\)0-110-1-111-14-12000
\(e_{11}\)-22201-2201-14-1-1-11
\(e_{12}\)2-1-10-100102-1401-1
\(e_{13}\)01-1-200-2-110-10401
\(e_{14}\)20-11000120-11040
\(e_{15}\)-221-12-11-1001-1104
 
[[4, -2, -2, 2, -2, 1, -1, 2, 1, 0, -2, 2, 0, 2, -2], [-2, 4, 2, -2, 2, -1, 1, -2, 1, -1, 2, -1, 1, 0, 2], [-2, 2, 4, 0, 0, -2, 2, 0, 0, 1, 2, -1, -1, -1, 1], [2, -2, 0, 4, -2, -1, 1, 2, 1, 0, 0, 0, -2, 1, -1], [-2, 2, 0, -2, 4, 1, 1, -2, -1, -1, 1, -1, 0, 0, 2], [1, -1, -2, -1, 1, 4, -2, -1, -2, -1, -2, 0, 0, 0, -1], [-1, 1, 2, 1, 1, -2, 4, 1, 0, 1, 2, 0, -2, 0, 1], [2, -2, 0, 2, -2, -1, 1, 4, 1, 1, 0, 1, -1, 1, -1], [1, 1, 0, 1, -1, -2, 0, 1, 4, -1, 1, 0, 1, 2, 0], [0, -1, 1, 0, -1, -1, 1, 1, -1, 4, -1, 2, 0, 0, 0], [-2, 2, 2, 0, 1, -2, 2, 0, 1, -1, 4, -1, -1, -1, 1], [2, -1, -1, 0, -1, 0, 0, 1, 0, 2, -1, 4, 0, 1, -1], [0, 1, -1, -2, 0, 0, -2, -1, 1, 0, -1, 0, 4, 0, 1], [2, 0, -1, 1, 0, 0, 0, 1, 2, 0, -1, 1, 0, 4, 0], [-2, 2, 1, -1, 2, -1, 1, -1, 0, 0, 1, -1, 1, 0, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/4)^{5}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)1532
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 14)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 14)\, L^\Sigma(s - 7, \chi_{-1024n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number1630

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 1630q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots14
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 4 \rangle^{7}\)
Rank of \(R(L)\)7
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)no: \(M / R(L) \cong (\mathbb{Z}/2)^{2}\) for \(M = L \cap (R(L) \otimes \mathbb{Q})\)

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\(\cong \langle 4 \rangle\)
\(\Phi_{3}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)
\(\Phi_{4}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)
\(\Phi_{5}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)
\(\Phi_{6}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)
\(\Phi_{7}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)

Simple roots

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

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{6} + e_{7}\)\(4\)
\(\alpha^{(2)}_{1}\)\(e_{2} - e_{3} + e_{10} - e_{12}\)\(4\)
\(\alpha^{(3)}_{1}\)\(e_{2} - e_{3} + e_{4}\)\(4\)
\(\alpha^{(4)}_{1}\)\(2e_{2} - e_{3} + e_{4} + e_{5} - e_{7} + e_{8} + e_{9} + e_{10} - e_{11} - e_{13} - 2e_{14}\)\(4\)
\(\alpha^{(5)}_{1}\)\(e_{1} - e_{2} - e_{4} - e_{6} + e_{7} - e_{8}\)\(4\)
\(\alpha^{(6)}_{1}\)\(e_{1} + e_{3} - e_{8}\)\(4\)
\(\alpha^{(7)}_{1}\)\(3e_{1} - 3e_{2} + e_{3} - 3e_{4} - e_{5} + e_{7} - 2e_{8} + e_{9} + e_{11} - e_{12} - e_{13} + 2e_{15}\)\(4\)

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 15 and has no vector of norm \(-2\); this record is \(\Lambda_G(-1)\) for the group \(\mathfrak{G}_{10} \cong D_8\) of Table 10.2 of Hashimoto, whose invariant lattice \(\Lambda^G\) is in row 10 of Table 10.3.

The Gram tensor is the LLL reduction of the coinvariant lattice lattices[9,4] of Höhn and Mason: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 9 that is number 4 of rank 9 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 10 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.

  • 029Q \(U \oplus \langle 4 \rangle^{2} \oplus \langle -4 \rangle^{3}\) A lattice \(M\) with \(M \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{10}\) 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 \(\mathfrak{G}_{10}\) 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[9,4].