jaxvacua.util.orthogonal_lattice

orthogonal_lattice(gens_in)

Returns generators of the integer lattice orthogonal to the lattice spanned by gens_in.

Details

Given \(d\) generators \(g_1,\ldots,g_d \in \mathbb{Z}^n\) with \(d < n\), the function computes generators of the orthogonal complement lattice

\[L^\perp = \bigl\{ v \in \mathbb{Z}^n \;:\; v \cdot g_i = 0 \;\;\forall\, i=1,\ldots,d \bigr\} \,,\]

which has rank \(n - d\).

The algorithm constructs the augmented matrix

\[\begin{split}B = \begin{pmatrix} c\,G \\ I_n \end{pmatrix} \in \mathbb{Z}^{(d+n)\times n} \,,\end{split}\]

where \(G\) is the \(d\times n\) generator matrix and \(c\) is an integer scale chosen so that LLL reduction on \(B^T\) separates the null-space rows. The last \(n-d\) rows of the LLL-reduced matrix (extracted from the \(I_n\) block) are the desired generators. The LLL computation uses flint.fmpz_mat for exact integer arithmetic.

Parameters:

gens_in (List[List[int]]) – List of \(d\) integer generator vectors of length \(n\), with \(d < n\).

Returns:

• list – List of \(n-d\) integer generators of \(L^\perp\),

• each of length :math:`n`.

Return type:

List[List[int]]

See also: extended_euclidean(), and the conifold-specific jaxvacua.conifold.conifold_utils.get_basis_change().