\(E_{8} \to \Lambda_{24}\)
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[16,1] of Höhn and
Mason, arXiv:1505.06420, in the Leech lattice.
latticedb check computes this map again from
sources/hoehn_mason/lattices_16_1.json with
latticedb.hoehn_mason.embedding.
\(E_{8}(2) \hookrightarrow \Lambda_{24}\)
A morphism \(E_{8}(2) \to \Lambda_{24}\): \(b_T(\varphi x, \varphi y) = 2 \, b_S(x, y)\).
The coinvariant lattice \(A\) of lattices[16,1] of Höhn and Mason, arXiv:1505.06420: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 16.
\[\left(\begin{array}{rrrrrrrr}
-4 & 3 & 2 & 2 & -4 & 1 & -1 & 0 \\
0 & -1 & 0 & 0 & 1 & -1 & 1 & -1 \\
1 & -1 & 0 & -1 & 2 & -1 & 0 & 0 \\
1 & 0 & 0 & -1 & 1 & -1 & 1 & 0 \\
1 & -1 & -2 & 0 & 1 & 1 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 \\
1 & 0 & -1 & 0 & 0 & 1 & -1 & 1 \\
2 & -2 & -2 & 1 & 0 & 1 & -1 & 1 \\
1 & -1 & 0 & 0 & 1 & -1 & 0 & 0 \\
1 & 0 & 0 & -1 & 1 & -1 & 1 & 0 \\
-1 & 0 & -1 & 1 & -1 & 2 & -1 & 0 \\
1 & -1 & -1 & 0 & 0 & 2 & -1 & 0 \\
0 & 1 & 1 & -1 & 0 & -1 & 1 & 0 \\
-1 & 0 & 1 & 1 & -1 & -1 & 1 & -1 \\
0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 \\
1 & -2 & -1 & 1 & 0 & 0 & 0 & 1 \\
-1 & 1 & 1 & -1 & 0 & 0 & 1 & -1 \\
-1 & 0 & 1 & 0 & 0 & 0 & 0 & -1 \\
0 & 1 & 1 & -1 & 0 & 0 & 0 & 0 \\
-1 & 1 & 1 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & -1 & -1 & 1 & 0 & 0 & 0 \\
0 & -2 & -1 & 2 & -1 & 1 & -1 & 1 \\
\end{array}\right)\]
{"matrix": [
[-4, 3, 2, 2, -4, 1, -1, 0],
[0, -1, 0, 0, 1, -1, 1, -1],
[1, -1, 0, -1, 2, -1, 0, 0],
[1, 0, 0, -1, 1, -1, 1, 0],
[1, -1, -2, 0, 1, 1, -1, 1],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, -1, 1, 0, 0, 0],
[1, 0, -1, 0, 0, 1, -1, 1],
[2, -2, -2, 1, 0, 1, -1, 1],
[1, -1, 0, 0, 1, -1, 0, 0],
[1, 0, 0, -1, 1, -1, 1, 0],
[-1, 0, -1, 1, -1, 2, -1, 0],
[1, -1, -1, 0, 0, 2, -1, 0],
[0, 1, 1, -1, 0, -1, 1, 0],
[-1, 0, 1, 1, -1, -1, 1, -1],
[0, 0, 0, 0, 1, -2, 1, 0],
[1, -2, -1, 1, 0, 0, 0, 1],
[-1, 1, 1, -1, 0, 0, 1, -1],
[-1, 0, 1, 0, 0, 0, 0, -1],
[0, 1, 1, -1, 0, 0, 0, 0],
[-1, 1, 1, 0, -1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0],
[1, 1, -1, -1, 1, 0, 0, 0],
[0, -2, -1, 2, -1, 1, -1, 1]
],
"row_subdivisions": [],
"column_subdivisions": []}