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Motivic Splittings For Symmetric Matrices

Anubhav Nanavaty

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 $p$.

Motivic Splittings For Symmetric Matrices

Abstract

We show that the space of symmetric matrices of a fixed rank over a field of characteristic not equal to 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 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 are split Tate in Voevodsky's category of motives in characteristic and Kelly's category of motives in characteristic .

Paper Structure

This paper contains 3 sections, 29 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Computations in the Grothendieck Ring
  3. Computing the Motive

Key Result

Theorem 1.1

Let $K$ be a field of characteristic not equal to $2$. In $C^{gm}$, for all $n,k,\ell \geq 0$, the Voevodsky motives $M_{gm}(\mathrm{Sym}^{n, k})$ and $M_{gm}^c(\mathrm{Sym}^{n, [k,\ell]})$ are split Tate. In other words, they are direct sums of the Tate motives $\mathbb{F}(i)[j]$, for $i,j\in\mathb

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • ...and 42 more