Rook Theory of the Etzion-Silberstein Conjecture
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 -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 -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.
Keywords
Cite
@article{arxiv.2209.05114,
title = {Rook Theory of the Etzion-Silberstein Conjecture},
author = {Anina Gruica and Alberto Ravagnani},
journal= {arXiv preprint arXiv:2209.05114},
year = {2022}
}