Motivic Splittings For Symmetric Matrices

TL;DR

This paper proves that the space of symmetric matrices of fixed rank over fields of characteristic not 2 is split Tate, using a field-independent filtration and point-counting techniques, with implications in algebraic geometry and motives.

Contribution

It introduces a novel filtration of symmetric matrices of fixed rank that is independent of the field, enabling the proof that these spaces are split Tate in various categories of motives.

Findings

The space of symmetric matrices of fixed rank is split Tate.

The filtration allows explicit computation of isomorphism classes in the Grothendieck ring.

The results hold over fields of characteristic not equal to 2 in both characteristic 0 and p.

Abstract

We show that the space of symmetric matrices of a fixed rank $k$ over a field $K$ of characteristic not equal to $2$ is split Tate. We do this by promoting the point-counting strategy of MacWilliams over finite fields to a filtration of the locus of rank $\leq k$ symmetric matrices that is independent of the field. This filtration immediately allows for a computation of their isomorphism classes in the Grothendieck ring of varieties in terms of the Lefschetz motive. We then promote this computation to prove that the space of symmetric matrices of a fixed rank $k$ are split Tate in Voevodsky's category of motives in characteristic $0$ and Kelly's category of motives in characteristic $p$.