\(\Lambda_{\mathfrak{G}_{32}}(-1) \to \Lambda_{\mathfrak{G}_{32}}(-1)\)
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[6,21] 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
\(\mathfrak{G}_{32} \cong
\mathrm{Hol}(C_5)\) of symplectic isometries.
latticedb check computes these maps again from
sources/hoehn_mason/lattices_6_21.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[6,21] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 20, the order of \(\mathfrak{G}_{32} \cong \mathrm{Hol}(C_5)\) of Table 10.2 of Hashimoto.
\[\left(\begin{array}{rrrrrrrrrrrrrrrrrr}
2 & 1 & 1 & -1 & 0 & -1 & -1 & -2 & 0 & -1 & -1 & 0 & -3 & -2 & -4 & 0 & 1 & 1 \\
-1 & 0 & 0 & 2 & 1 & 1 & 0 & 0 & -1 & 0 & 3 & -1 & 0 & -2 & -1 & -1 & 2 & 1 \\
0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & -1 & -1 & 2 & 0 & -1 & -2 & -3 & -1 & 2 & 2 \\
1 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & -2 & -2 & 0 & 0 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & -2 & -1 & 1 & 1 \\
0 & 1 & 2 & 0 & -1 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & -2 & -1 & 1 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & -1 & -1 & 1 & 1 \\
1 & 0 & -1 & -1 & -1 & 0 & 0 & 0 & 1 & 0 & -2 & 1 & 0 & 1 & 1 & 1 & -2 & 0 \\
-1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & -1 & 1 & 0 \\
1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & 1 & -1 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & -1 & 0 & 1 & 1 \\
-1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & 1 & 0 \\
0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
1 & 0 & -1 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & -1 & 1 & 0 & 1 \\
-1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 2 & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\
0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 1 & 0 & 0 & -1 & -1 \\
0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
\end{array}\right)\]
{"matrix": [
[2, 1, 1, -1, 0, -1, -1, -2, 0, -1, -1, 0, -3, -2, -4, 0, 1, 1],
[-1, 0, 0, 2, 1, 1, 0, 0, -1, 0, 3, -1, 0, -2, -1, -1, 2, 1],
[0, 0, 0, 0, 2, 0, 0, 0, -1, -1, 2, 0, -1, -2, -3, -1, 2, 2],
[1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, -2, -2, 0, 0, 1],
[0, 0, 1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, -2, -1, 1, 1],
[0, 1, 2, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -2, -1, 1, 0],
[0, 0, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, -1, 1, 1],
[1, 0, -1, -1, -1, 0, 0, 0, 1, 0, -2, 1, 0, 1, 1, 1, -2, 0],
[-1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, -1, 1, 0],
[1, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, -1, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, 0, 1, 1],
[-1, 0, 1, 1, 0, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, -1, 1, 0],
[0, 0, 1, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1],
[1, 0, -1, 0, 1, 0, 0, -1, 0, 0, 0, 0, -1, -1, -1, 1, 0, 1],
[-1, 0, 0, 0, 1, 0, 0, 1, -1, 0, 2, 0, 0, 0, 0, -1, 1, 0],
[0, 0, 1, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 1, 0, 0, -1, -1],
[0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 0, 1, 0]
],
"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[6,21] of Höhn and Mason, arXiv:1505.06420, on the coinvariant lattice. The group \(C\) has order 20, the order of \(\mathfrak{G}_{32} \cong \mathrm{Hol}(C_5)\) of Table 10.2 of Hashimoto.
\[\left(\begin{array}{rrrrrrrrrrrrrrrrrr}
0 & 2 & 0 & 0 & -1 & -2 & 1 & 2 & 0 & 1 & 0 & 2 & 1 & 0 & 1 & -2 & 3 & -3 \\
0 & 0 & -2 & 1 & 2 & -1 & 0 & 2 & -1 & 1 & 2 & 1 & 0 & -2 & 0 & -1 & 3 & 0 \\
0 & 2 & -1 & 0 & 0 & -2 & 0 & 2 & 0 & 2 & 2 & 2 & 1 & 0 & 1 & -2 & 3 & -2 \\
0 & 1 & 0 & 0 & 0 & -1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 2 & -1 \\
0 & 1 & 0 & 1 & -1 & -1 & 0 & 2 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & -1 & 2 & -2 \\
0 & 1 & 1 & 0 & -1 & -2 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & -1 & 1 & -3 \\
0 & 0 & 0 & -1 & 1 & 0 & 0 & -1 & -1 & 0 & 1 & 0 & -1 & 0 & -1 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & -1 \\
0 & -1 & 1 & 0 & -1 & 1 & 0 & -1 & 1 & -1 & -2 & -1 & 1 & 1 & 0 & 1 & -2 & 1 \\
0 & 1 & 1 & -1 & 0 & -1 & 0 & 0 & 0 & 1 & 1 & 0 & -1 & 1 & 0 & -1 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & -1 & 1 & -1 \\
0 & 0 & -1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 2 & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\
0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 1 & 0 & 0 & -1 & -1 \\
0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
\end{array}\right)\]
{"matrix": [
[0, 2, 0, 0, -1, -2, 1, 2, 0, 1, 0, 2, 1, 0, 1, -2, 3, -3],
[0, 0, -2, 1, 2, -1, 0, 2, -1, 1, 2, 1, 0, -2, 0, -1, 3, 0],
[0, 2, -1, 0, 0, -2, 0, 2, 0, 2, 2, 2, 1, 0, 1, -2, 3, -2],
[0, 1, 0, 0, 0, -1, 1, 1, 0, 0, 0, 1, 0, 0, 0, -1, 2, -1],
[0, 1, 0, 1, -1, -1, 0, 2, 1, 1, 0, 1, 1, 0, 1, -1, 2, -2],
[0, 1, 1, 0, -1, -2, 0, 1, 0, 1, 0, 1, 0, 1, 1, -1, 1, -3],
[0, 0, 0, -1, 1, 0, 0, -1, -1, 0, 1, 0, -1, 0, -1, 0, 0, 1],
[0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 1, 0, 0, 0, 0, -1, 1, -1],
[0, -1, 1, 0, -1, 1, 0, -1, 1, -1, -2, -1, 1, 1, 0, 1, -2, 1],
[0, 1, 1, -1, 0, -1, 0, 0, 0, 1, 1, 0, -1, 1, 0, -1, 0, -1],
[0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1],
[0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0, -1, 1, -1],
[0, 0, -1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0],
[0, 0, 1, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1],
[0, 0, -1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1, 0, 1, 0],
[-1, 0, 0, 0, 1, 0, 0, 1, -1, 0, 2, 0, 0, 0, 0, -1, 1, 0],
[0, 0, 1, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 1, 0, 0, -1, -1],
[0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 0, 1, 0]
],
"row_subdivisions": [],
"column_subdivisions": []}