\(\Lambda_{\mathfrak{G}_{78}}(-1) \to \Lambda_{24}\)
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 coinvariant lattice of lattices[5,9] of Höhn and
Mason, arXiv:1505.06420, in the Leech lattice.
latticedb check computes this map again from
sources/hoehn_mason/lattices_5_9.json with
latticedb.hoehn_mason.embedding.
\(\Lambda_{\mathfrak{G}_{78}}(-1) \hookrightarrow \Lambda_{24}\)
The coinvariant lattice \(A\) of lattices[5,9] of Höhn and Mason, arXiv:1505.06420: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 5.
\[\left(\begin{array}{rrrrrrrrrrrrrrrrrrr}
-4 & -1 & -4 & -4 & 3 & 7 & 2 & -2 & 0 & -4 & -2 & 3 & 0 & -4 & 3 & 3 & -4 & 0 & -7 \\
2 & 1 & 2 & 1 & 0 & -2 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 1 & -1 & -1 & 1 & -1 & 2 \\
2 & 1 & 2 & 2 & 0 & -3 & 0 & 1 & -1 & 1 & 0 & -1 & -1 & 1 & -1 & -1 & 2 & 0 & 3 \\
1 & 1 & 1 & 2 & -1 & -2 & 0 & 1 & 0 & 2 & 1 & -1 & -1 & 1 & -1 & -1 & 0 & 0 & 2 \\
0 & 0 & 1 & 1 & -1 & -1 & -1 & 1 & 0 & 2 & 0 & 0 & 1 & 1 & 0 & -1 & 1 & 1 & 2 \\
1 & 1 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 1 & 0 & -1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & -1 & 1 & 0 & 0 & -1 & 0 & 1 & -1 & 0 & 0 & 1 \\
-1 & -2 & -1 & -1 & -1 & 0 & -2 & 1 & 1 & -1 & -1 & 1 & 1 & 0 & -1 & 1 & 1 & 1 & -1 \\
-1 & -2 & 0 & -1 & -1 & -1 & -1 & 1 & 0 & 1 & 0 & -1 & 1 & 1 & -1 & 1 & 2 & 1 & 1 \\
1 & -1 & 1 & -1 & -1 & -2 & -1 & 0 & 0 & -1 & 0 & -1 & 1 & 1 & -2 & 1 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & -1 & 1 & 0 & -1 & -1 & 0 & -1 & 0 & 1 & 0 & 2 \\
-1 & 0 & -1 & 0 & 0 & 2 & 0 & -1 & 1 & 0 & 0 & 1 & 1 & 0 & 2 & -1 & -1 & 0 & -2 \\
-1 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & -1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & -1 & -1 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1 & 0 & 1 & 0 & 0 & 0 & -1 & -1 & -1 \\
2 & 1 & 1 & 1 & 0 & -1 & -1 & 0 & 0 & -1 & 0 & 0 & -1 & 1 & -1 & -1 & 0 & -1 & 1 \\
0 & -1 & 0 & 0 & -1 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & -1 & 0 & 1 & -1 & -1 & 0 & 0 \\
0 & 0 & 0 & -1 & 1 & 0 & 1 & -1 & -1 & -1 & -1 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & -1 \\
1 & 1 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & 0 & 1 \\
-1 & 0 & -1 & -2 & 1 & 2 & 1 & -1 & -1 & -1 & 0 & 0 & 0 & -1 & 1 & 1 & -1 & 0 & -1 \\
1 & 0 & 1 & 1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 1 & 0 & 0 \\
1 & 1 & 0 & 2 & -1 & -1 & -1 & 1 & 1 & 0 & 0 & 1 & -1 & 1 & 0 & -1 & 0 & 0 & 1 \\
-3 & -2 & -1 & -3 & 1 & 2 & 1 & 0 & -1 & 1 & -1 & -1 & 1 & -1 & 0 & 2 & 0 & 1 & -1 \\
\end{array}\right)\]
{"matrix": [
[-4, -1, -4, -4, 3, 7, 2, -2, 0, -4, -2, 3, 0, -4, 3, 3, -4, 0, -7],
[2, 1, 2, 1, 0, -2, 0, 0, 0, 1, 1, -1, 0, 1, -1, -1, 1, -1, 2],
[2, 1, 2, 2, 0, -3, 0, 1, -1, 1, 0, -1, -1, 1, -1, -1, 2, 0, 3],
[1, 1, 1, 2, -1, -2, 0, 1, 0, 2, 1, -1, -1, 1, -1, -1, 0, 0, 2],
[0, 0, 1, 1, -1, -1, -1, 1, 0, 2, 0, 0, 1, 1, 0, -1, 1, 1, 2],
[1, 1, 1, 1, 0, -1, 0, 0, 0, 1, 1, -1, 0, 1, 0, -1, 0, 0, 1],
[0, 1, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, -1, 0, 1, -1, 0, 0, 1],
[-1, -2, -1, -1, -1, 0, -2, 1, 1, -1, -1, 1, 1, 0, -1, 1, 1, 1, -1],
[-1, -2, 0, -1, -1, -1, -1, 1, 0, 1, 0, -1, 1, 1, -1, 1, 2, 1, 1],
[1, -1, 1, -1, -1, -2, -1, 0, 0, -1, 0, -1, 1, 1, -2, 1, 1, 0, 1],
[0, 0, 0, 0, 0, -1, 0, 1, -1, 1, 0, -1, -1, 0, -1, 0, 1, 0, 2],
[-1, 0, -1, 0, 0, 2, 0, -1, 1, 0, 0, 1, 1, 0, 2, -1, -1, 0, -2],
[-1, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, 0, 1, 1, 0],
[0, 1, -1, 1, 0, 1, 1, 0, 0, 0, 0, 1, -1, -1, 1, 0, -1, -1, 0],
[1, 0, 1, 0, 0, 0, 0, -1, 1, -1, 1, 0, 1, 0, 0, 0, -1, -1, -1],
[2, 1, 1, 1, 0, -1, -1, 0, 0, -1, 0, 0, -1, 1, -1, -1, 0, -1, 1],
[0, -1, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0],
[0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, -1, 0, 1, -1, -1, 0, 0],
[0, 0, 0, -1, 1, 0, 1, -1, -1, -1, -1, 0, 0, -1, 0, 1, 0, 0, -1],
[1, 1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1],
[-1, 0, -1, -2, 1, 2, 1, -1, -1, -1, 0, 0, 0, -1, 1, 1, -1, 0, -1],
[1, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0],
[1, 1, 0, 2, -1, -1, -1, 1, 1, 0, 0, 1, -1, 1, 0, -1, 0, 0, 1],
[-3, -2, -1, -3, 1, 2, 1, 0, -1, 1, -1, -1, 1, -1, 0, 2, 0, 1, -1]
],
"row_subdivisions": [],
"column_subdivisions": []}