The lattice generated by a sequence of elements of a rational vector space

[Michel VAN DEN BERGH]

If somebody could help me with the following problem I would appreciate it!

Let V=QQ^n be an n-dimensional vector space over the rationals, equipped with a symmetric bilinear form B (given by its evaluation on the standard basis vectors of QQ^n).

Assume given v1,..,vm in V. I would like to construct the IntegralLattice spanned

by the (vi)_i, equipped with the restriction of the bilinear form B. If it helps: v1, ..., vm generate V.

Any ideas on how to do this in a nice way?

[Nils Bruin]

You can get Z^m with the bilinear form on it simply by computing the Gram matrix G of the pairing relative to v1,...,vm.

If your pairing is non-degenerate, the lattice you're looking for is Z^m/ker(G). and the bilinear form can be induced on that quotient from the information that you have in G already.

(you may well have to clear denominators from G in order to get sage to compute ker(G) for you)

If your pairing is degenerate then getting the Z^n -lattice that v1,...,vm span needs to use the actual embedding in V. In that case, you can just scale out the denominators of the coordinates of your v1,...,vm and compute a Hermite normal form of the resulting integer matrix. That's a scaling away from the actual lattice generated by v1,...,vm. Getting the Gram matrix on that representation is also straightforward.

These are not quite one-liners, but they do use fairly high-level linear algebra, so it shouldn't be too onerous to do.

Once you have the Gram matrix you can construct an "IntegralLattice" from that.

[Michel VAN DEN BERGH]

Thanks! That makes sense. I will try!