Joel Brewster Lewis: Counting matrices over finite fields with restricted entries
Abstract
Classical formulas show that the number of invertible n-by-n matrices over a finite field with q
elements is a natural q-analogue of n!, and more generally that the number of n-by-n matrices of
rank r is a q-analogue of the number of ways to place r nonattacking rooks on an n-by-n board. In
this talk, we study the functions that count matrices of given rank over a finite field with
specified positions equal to 0. We show that the number invertible matrices with zero diagonal is a
natural q-analogue of the number of derangements (i.e., permutations with no fixed points). More
generally, we show that the number of matrices of given rank with certain entries equal to 0 is a
q-analogue of rook placements with restricted positions.
In addition, we study the question of when the number of matrices with given size, rank, and
prescribed entries equal to 0 is a polynomial in the size q of the field. Generalizing work of
Stembridge and Haglund, we give a variety of structural conditions for polynomiality. In
particular, we show that the number of matrices whose support lies in a skew shape is a polynomial
with positive coefficients. We also study the situation in which the prescribed 0s are the entries
of the Rothe diagram of a permutation, and give several intriguing conjectures in this case.
This work is joint with Aaron Klein, Ricky Liu, Alejandro Morales, Greta
Panova, Steven Sam and Yan Zhang.