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arXiv · 2209.05114

Rook Theory of the Etzion-Silberstein Conjecture

Anina Gruica·Alberto Ravagnani
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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.

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Anina Gruica, Alberto Ravagnani. 2022-09-12. Rook Theory of the Etzion-Silberstein Conjecture. https://arxiv.org/abs/2209.05114

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