Rook Theory of the Etzion-Silberstein Conjecture

TL;DR

This paper links the Etzion-Silberstein Conjecture on matrix spaces with rook theory, providing new instances and solving its asymptotic version over large finite fields through combinatorial methods.

Contribution

It establishes a connection between the conjecture and $q$-rook polynomials, offering new cases and an asymptotic solution using combinatorial techniques.

Findings

Connected the conjecture to $q$-rook polynomials.

Provided a closed formula for the trailing degree of $q$-rook polynomials.

Solved the asymptotic version of the conjecture over large finite fields.

Abstract

In 2009, Etzion and Siberstein proposed a conjecture on the largest dimension of a linear space of matrices over a finite field in which all nonzero matrices are supported on a Ferrers diagram and have rank bounded below by a given integer. Although several cases of the conjecture have been established in the past decade, proving or disproving it remains to date a wide open problem. In this paper, we take a new look at the Etzion-Siberstein Conjecture, investigating its connection with rook theory. Our results show that the combinatorics behind this open problem is closely linked to the theory of $q$-rook polynomials associated with Ferrers diagrams, as defined by Garsia and Remmel. In passing, we give a closed formula for the trailing degree of the $q$-rook polynomial associated with a Ferrers diagram in terms of the cardinalities of its diagonals. The combinatorial approach taken in this paper allows us to establish some new instances of the Etzion-Silberstein Conjecture using a non-constructive argument. We also solve the asymptotic version of the conjecture over large finite fields, answering a current open question.