\(U^{3} \oplus E_{8}(-2) \to U^{3} \oplus E_{8}(-1)^{2}\)
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 invariant and coinvariant lattices of the finite symplectic
groups of Hashimoto in the K3 lattice. The anti-isometry maps the
generators of the normal form of SageMath of one discriminant form to
those of the other twisted by \(-1\);
three hyperbolic planes are split off and qfisom of PARI/GP
maps the remainder onto \(E_8(-1)^2\).
\(U^{3} \oplus E_{8}(-2) = \Lambda^G \hookrightarrow \mathrm{II}_{3,19}\), row 1
The invariant lattice \(\Lambda^G\) of \(\mathfrak{G}_{1} \cong C_2\) of Table 10.2 of Hashimoto, with orthogonal complement the image of \(E_{8}(-2)\). The orthogonal sum \(U^{3} \oplus E_{8}(-2) \oplus E_{8}(-2)\) glued along an anti-isometry of the discriminant forms is even, unimodular and of signature \((3, 19)\); its isometry onto the basis of \(\mathrm{II}_{3,19}\) restricts to this map.
\[\left(\begin{array}{rrrrrrrrrrrrrr}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -2 & -2 & -2 & 0 & 0 & 2 & 2 & -2 & 0 & -2 & 2 & 0 \\
0 & 0 & -1 & -1 & -1 & 0 & 0 & 1 & 1 & -1 & 0 & -1 & 1 & 0 \\
0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & -1 & -1 & 1 & 0 & -1 & 1 & 0 & 0 & -1 & 0 & 1 \\
0 & 0 & -1 & -1 & -1 & 0 & 0 & -1 & 2 & -1 & 0 & -1 & 0 & 2 \\
0 & 0 & -1 & -1 & -1 & 0 & 0 & -2 & 2 & 0 & 0 & -2 & 0 & 2 \\
0 & 0 & -2 & -2 & -2 & 0 & 0 & -2 & 4 & -2 & 1 & -3 & 0 & 3 \\
0 & 0 & -2 & -2 & -2 & 0 & 0 & -1 & 4 & -2 & 0 & -2 & 0 & 2 \\
0 & 0 & -2 & -2 & -2 & 0 & -1 & 0 & 4 & -2 & 0 & -2 & 0 & 2 \\
0 & 0 & -2 & -2 & -2 & 0 & 0 & 1 & 3 & -2 & 0 & -2 & 0 & 2 \\
0 & 0 & -1 & -1 & -1 & 0 & 0 & 0 & 2 & -1 & 0 & -1 & 0 & 1 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & -2 & 1 & 0 & 2 & -2 & 1 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & -3 & 2 & 0 & 2 & -2 & 0 \\
0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & -4 & 2 & 1 & 3 & -4 & 1 \\
0 & 0 & 3 & 3 & 3 & 0 & 0 & 0 & -6 & 4 & 0 & 5 & -6 & 1 \\
0 & 0 & 2 & 2 & 2 & 0 & 0 & 1 & -4 & 2 & 0 & 4 & -4 & 0 \\
0 & 0 & 2 & 2 & 2 & 0 & 1 & 0 & -4 & 2 & 0 & 4 & -4 & 0 \\
0 & 0 & 2 & 2 & 2 & 0 & 0 & -1 & -3 & 2 & 0 & 4 & -4 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & -2 & 1 & 0 & 2 & -2 & 0 \\
\end{array}\right)\]
{"matrix": [
[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, -2, -2, -2, 0, 0, 2, 2, -2, 0, -2, 2, 0],
[0, 0, -1, -1, -1, 0, 0, 1, 1, -1, 0, -1, 1, 0],
[0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, -1, -1, -1, 1, 0, -1, 1, 0, 0, -1, 0, 1],
[0, 0, -1, -1, -1, 0, 0, -1, 2, -1, 0, -1, 0, 2],
[0, 0, -1, -1, -1, 0, 0, -2, 2, 0, 0, -2, 0, 2],
[0, 0, -2, -2, -2, 0, 0, -2, 4, -2, 1, -3, 0, 3],
[0, 0, -2, -2, -2, 0, 0, -1, 4, -2, 0, -2, 0, 2],
[0, 0, -2, -2, -2, 0, -1, 0, 4, -2, 0, -2, 0, 2],
[0, 0, -2, -2, -2, 0, 0, 1, 3, -2, 0, -2, 0, 2],
[0, 0, -1, -1, -1, 0, 0, 0, 2, -1, 0, -1, 0, 1],
[0, 0, 1, 1, 1, 0, 0, 0, -2, 1, 0, 2, -2, 1],
[0, 0, 1, 1, 1, 0, 0, 0, -3, 2, 0, 2, -2, 0],
[0, 0, 2, 2, 2, 0, 0, 0, -4, 2, 1, 3, -4, 1],
[0, 0, 3, 3, 3, 0, 0, 0, -6, 4, 0, 5, -6, 1],
[0, 0, 2, 2, 2, 0, 0, 1, -4, 2, 0, 4, -4, 0],
[0, 0, 2, 2, 2, 0, 1, 0, -4, 2, 0, 4, -4, 0],
[0, 0, 2, 2, 2, 0, 0, -1, -3, 2, 0, 4, -4, 0],
[0, 0, 1, 1, 1, 0, 0, 0, -2, 1, 0, 2, -2, 0]
],
"row_subdivisions": [],
"column_subdivisions": []}
Diagonal into the summands \(\{1\} \sqcup \{2\} \sqcup \{3\} \sqcup \{4, 5\}\)
Summand \(j\) of the source maps by \(x \mapsto (x, \ldots, x)\) into the orthogonal summands of the target in set \(j\), which are counted from 1 in the order of the basis.
\[\left(\begin{array}{rrrrrrrrrrrrrr}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}\right)\]
{"matrix": [
[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, 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, 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, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
[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, 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": []}