Morphisms 0149 \(\to\) 0149

\(\Lambda_{12} \to \Lambda_{12}\)

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[12,1] 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 \(\mathfrak{G}_{3} \cong C_2^2\) of symplectic isometries. latticedb check computes these maps again from sources/hoehn_mason/lattices_12_1.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[12,1] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 4, the order of \(\mathfrak{G}_{3} \cong C_2^2\) of Table 10.2 of Hashimoto.

\[\left(\begin{array}{rrrrrrrrrrrr} 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ \end{array}\right)\]
{"matrix": [
  [1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0],
  [0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 1, 0, -1, 1, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0],
  [0, 0, 0, 0, 0, 0, 0, 0, 0, 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[12,1] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 4, the order of \(\mathfrak{G}_{3} \cong C_2^2\) of Table 10.2 of Hashimoto.

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