On affine spaces of alternating matrices with constant rank

TL;DR

This paper extends the understanding of the maximum dimension of affine subspaces of alternating matrices with constant rank over large fields, generalizing previous real-field results and identifying key structural properties.

Contribution

It generalizes Rubei's real-field results to all sufficiently large fields and characterizes the largest affine subspaces of constant rank matrices in alternating matrix spaces.

Findings

Determined maximum dimension of affine subspaces with constant rank over large fields.

Extended results from real numbers to arbitrary fields with large enough cardinality.

Identified challenges for cases where n ≤ r+2.

Abstract

Let $\mathbb{F}$ be a field, and $n \geq r>0$ be integers, with $r$ even. Denote by $\mathrm{A}_n(\mathbb{F})$ the space of all $n$-by-$n$ alternating matrices with entries in $\mathbb{F}$. We consider the problem of determining the greatest possible dimension for an affine subspace of $\mathrm{A}_n(\mathbb{F})$ in which every matrix has rank equal to $r$ (or rank at least $r$). Recently Rubei has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that $n \geq r+3$ and $|\mathbb{F}|\geq \min\bigl(r-1,\frac{r}{2}+2\bigr)$, we also determine the affine subspaces of rank $r$ matrices in $\mathrm{A}_n(\mathbb{F})$ that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case $n\leq r+2$.