Rook theory of the finite general linear group
Combinatorics
2021-03-30 v2 Information Theory
Algebraic Geometry
math.IT
Abstract
Matrices over a finite field having fixed rank and restricted support are a natural -analogue of rook placements on a board. We develop this -rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding theory, we show that these -hit and -rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier -analogues of rook theory, as well as settling a polynomiality conjecture and finding a counterexample of a positivity conjecture of the authors and Klein.
Keywords
Cite
@article{arxiv.1707.08192,
title = {Rook theory of the finite general linear group},
author = {Joel Brewster Lewis and Alejandro H. Morales},
journal= {arXiv preprint arXiv:1707.08192},
year = {2021}
}
Comments
25 pages, 10 figure files. Minor change in definition of q-hit numbers changes notation but doesn't substantively affect results