\(\Lambda_{\mathfrak{G}_{9}}(-1) \to \Lambda_{\mathfrak{G}_{9}}(-1)\)
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[10,1] 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
\(\mathfrak{G}_{9} \cong C_2^3\) of
symplectic isometries. latticedb check computes these maps
again from sources/hoehn_mason/lattices_10_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[10,1] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 8, the order of \(\mathfrak{G}_{9} \cong C_2^3\) of Table 10.2 of Hashimoto.
{"matrix": [
[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[-1, 0, 0, 1, 0, -1, 1, -1, 0, 0, 1, 0, -1, 1],
[0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1, -2],
[0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 1, -1, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, 2],
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, -2],
[1, -1, 0, -1, 1, 0, 0, 0, -1, 0, -1, 1, 1, -2],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1],
[0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0],
[0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, -1, 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, 0, 0, 1, 0],
[0, 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[10,1] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 8, the order of \(\mathfrak{G}_{9} \cong C_2^3\) of Table 10.2 of Hashimoto.
{"matrix": [
[-1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0],
[0, -1, 0, -1, 1, 0, 1, 0, 0, -1, -1, 0, 0, 0],
[0, 0, -1, 0, 0, 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, -1, 0],
[0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 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, 0, 0, 1, 0, 0],
[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, 0, 0, -1]
],
"row_subdivisions": [],
"column_subdivisions": []}
Generator 3 of the stabilizer of \(B\) in \(\mathrm{Co}_0\)
Generator 3 of the pointwise stabilizer \(C \subset \mathrm{Co}_0\) of the fixed-point lattice \(B\) of lattices[10,1] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 8, the order of \(\mathfrak{G}_{9} \cong C_2^3\) of Table 10.2 of Hashimoto.
{"matrix": [
[-1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0],
[0, 0, 0, 0, -1, 1, 0, -1, 0, -1, 0, 0, 0, 0],
[0, 0, -1, -1, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
[1, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 1, 1, 0],
[1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, 1, 0],
[-1, 1, 0, 0, -1, 0, -1, 1, 1, 0, 0, -1, -1, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[-1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, -1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, -1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
],
"row_subdivisions": [],
"column_subdivisions": []}