Morphisms 02BG \(\to\) 02BG

\(\Lambda_{\mathfrak{G}_{12}}(-1) \to \Lambda_{\mathfrak{G}_{12}}(-1)\)

Column \(j\) of each matrix gives \(\varphi(e_j)\) in the chosen basis of the target; the subdivisions separate orthogonal summands of the source (columns) and of the target (rows).

Notes

The generators of the pointwise stabilizer \(C\) of the fixed-point lattice \(B\) of lattices[7,8] of Höhn and Mason, arXiv:1505.06420, as automorphisms of the coinvariant lattice \(A = B^\perp\). Through the gluing of \(A(-1)\) and the invariant lattice into the K3 lattice, \(C\) is the group \(\mathfrak{G}_{12} \cong Q_8\) of symplectic isometries. latticedb check computes these maps again from sources/hoehn_mason/lattices_7_8.json with latticedb.hoehn_mason.automorphisms and the order of the group they generate.

Generator 1 of the stabilizer of \(B\) in \(\mathrm{Co}_0\)

Generator 1 of the pointwise stabilizer \(C \subset \mathrm{Co}_0\) of the fixed-point lattice \(B\) of lattices[7,8] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 8, the order of \(\mathfrak{G}_{12} \cong Q_8\) of Table 10.2 of Hashimoto.

\[\left(\begin{array}{rrrrrrrrrrrrrrrrr} 1 & 0 & -1 & -1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & -1 & 1 & 0 & -1 & 1 & 0 \\ 0 & -1 & 0 & -1 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 1 & 3 & 1 & 0 & 0 & 1 \\ 1 & 0 & -1 & 0 & -1 & 1 & 1 & 0 & 0 & 2 & 1 & -1 & -1 & 0 & 0 & 0 & -1 \\ 0 & -1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & -1 & 0 & 0 & -1 & -1 & 0 & -1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 & -1 & 0 & 0 & 0 & -1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 & -1 & 0 & 0 & -1 & -1 & 1 & 1 & 0 & -1 & 0 & 1 \\ -1 & 0 & 0 & 0 & 1 & -1 & -1 & 0 & 0 & -2 & -1 & 1 & 1 & 0 & 0 & -1 & 1 \\ 0 & -1 & 0 & 0 & 1 & -1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ \end{array}\right)\]
{"matrix": [
  [1, 0, -1, -1, 0, 1, 0, 0, 0, 1, 1, -1, 1, 0, -1, 1, 0],
  [0, -1, 0, -1, 1, -1, -1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 1],
  [1, 0, -1, 0, -1, 1, 1, 0, 0, 2, 1, -1, -1, 0, 0, 0, -1],
  [0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0],
  [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, 1, 0, 0, 0, 0],
  [0, 0, 0, 0, 1, -1, -1, 0, 0, -1, -1, 0, -1, 0, 0, 0, 1],
  [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1],
  [0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0],
  [0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0],
  [0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
  [0, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
  [0, 0, 0, -1, 1, 0, -1, 0, 0, 0, -1, 1, 1, 0, 0, 0, 0],
  [0, 0, 0, -1, 1, 0, -1, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1],
  [-1, 0, 0, 0, 1, -1, -1, 0, 0, -2, -1, 1, 1, 0, 0, -1, 1],
  [0, -1, 0, 0, 1, -1, -1, 0, 0, -1, 0, 0, 1, 1, 0, 0, 1]
],
 "row_subdivisions": [],
 "column_subdivisions": []}

Generator 2 of the stabilizer of \(B\) in \(\mathrm{Co}_0\)

Generator 2 of the pointwise stabilizer \(C \subset \mathrm{Co}_0\) of the fixed-point lattice \(B\) of lattices[7,8] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 8, the order of \(\mathfrak{G}_{12} \cong Q_8\) of Table 10.2 of Hashimoto.

\[\left(\begin{array}{rrrrrrrrrrrrrrrrr} -1 & 1 & 1 & 0 & -1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & -1 & 0 & 1 & -1 & -1 \\ -1 & 2 & -2 & 1 & -2 & 0 & -1 & 1 & -2 & 0 & 0 & 0 & -2 & -3 & 1 & -1 & 0 \\ -1 & 0 & 0 & 0 & 1 & -1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & -2 & 0 \\ 0 & 1 & -3 & 1 & -1 & 0 & -1 & 0 & -1 & 0 & -1 & -1 & -3 & -2 & 0 & -1 & 0 \\ 0 & 1 & -3 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & -1 & -1 & -1 & -2 & 0 & -1 & 0 \\ 0 & 0 & -1 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 1 & -1 & 0 \\ 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\ -1 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -2 & 0 & 0 & 0 & -1 & 0 & -1 & -1 & -1 & 0 & -1 & -1 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 & -1 & 1 & 0 & -1 & 0 & 1 & 0 & -1 & -1 & 0 & -1 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & -1 & 1 & 1 & -1 & 0 & 1 & 1 & -1 & -1 & 0 & 0 & 1 & -1 \\ 1 & 0 & -1 & 0 & -1 & 1 & 0 & -1 & -1 & 0 & 0 & -1 & -1 & 0 & -1 & 1 & 0 \\ 0 & 1 & -1 & 0 & -1 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 \\ \end{array}\right)\]
{"matrix": [
  [-1, 1, 1, 0, -1, 0, 0, 1, 1, 1, 1, 1, -1, 0, 1, -1, -1],
  [-1, 2, -2, 1, -2, 0, -1, 1, -2, 0, 0, 0, -2, -3, 1, -1, 0],
  [-1, 0, 0, 0, 1, -1, 0, 1, 1, 0, 0, 1, 1, 0, 1, -2, 0],
  [0, 1, -3, 1, -1, 0, -1, 0, -1, 0, -1, -1, -3, -2, 0, -1, 0],
  [0, 1, -3, 0, 0, 0, -1, 0, -1, 0, -1, -1, -1, -2, 0, -1, 0],
  [0, 0, -1, 1, 0, -1, 0, 0, 0, 0, 0, -1, -1, -1, 1, -1, 0],
  [0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0, 1, 1],
  [-1, 0, 2, 0, 0, -1, 0, 0, 0, -1, 0, 1, 1, 1, 0, 0, 1],
  [0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0],
  [0, 1, -2, 0, 0, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0],
  [0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
  [1, 0, -1, 0, -1, 1, 0, -1, 0, 1, 0, -1, -1, 0, -1, 1, -1],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0],
  [0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, 1, 0],
  [1, 0, 0, 0, -1, 1, 1, -1, 0, 1, 1, -1, -1, 0, 0, 1, -1],
  [1, 0, -1, 0, -1, 1, 0, -1, -1, 0, 0, -1, -1, 0, -1, 1, 0],
  [0, 1, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0]
],
 "row_subdivisions": [],
 "column_subdivisions": []}