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arXiv · 1707.08192

Rook theory of the finite general linear group

Joel Brewster Lewis·Alejandro H. Morales
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Abstract

Matrices over a finite field having fixed rank and restricted support are a natural $q$-analogue of rook placements on a board. We develop this $q$-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding theory, we show that these $q$-hit and $q$-rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier $q$-analogues of rook theory, as well as settling a polynomiality conjecture and finding a counterexample of a positivity conjecture of the authors and Klein.

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Joel Brewster Lewis, Alejandro H. Morales. 2017-09-05. Rook theory of the finite general linear group. https://doi.org/10.1080/10586458.2018.1470045

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