The rank 8 case of a conjecture on square-zero upper triangular matrices

TL;DR

This paper proves a specific case (rank 8) of a conjecture related to square-zero upper triangular matrices, with implications for group actions on products of spheres.

Contribution

It establishes the validity of a stronger conjecture for the case N=8, regardless of the field's characteristic, advancing understanding of matrix varieties and algebraic topology.

Findings

The stronger conjecture holds for N=8.

No characteristic restriction for the main result.

Implication for free actions of elementary abelian 2-groups on products of spheres.

Abstract

Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that any $\mathrm{dg}$-$A$-module that is free of rank $N$ as an $A$-module and whose homology is nontrivial and finite dimensional as a $k$-vector space satisfies $N\geq 2^r$. In this paper, we examine a stronger conjecture concerning varieties of square-zero upper triangular $N\times N$ matrices. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of rank $4$ cannot act freely on $X$.