University of Minnesota Combinatorics Seminar
Friday, September 30, 2011
3:35pm in 570 Vincent Hall



Counting matrices over finite fields

Steven Sam

MIT


Abstract

There are many classical formulas for the number of matrices of a given rank with entries in a finite field, and all of them are polynomial in the size of the field. These polynomials often have nice intepretations as q-analogues of objects such as permutations. I'll discuss the problem when we impose the restriction that certain entries of the matrix must be 0. It is not always true that such counting functions are polynomials, and I will discuss some partial results in this direction. One example is that polynomiality occurs when the diagonal is forced to be 0 and this gives a q-analogue of derangements. I'll also discuss a curious relationship between symmetric and skew-symmetric matrices which has an interpretation in terms of the R-polynomials of Kazhdan-Lusztig. Some of this is joint work with Liu--Lewis--Morales--Panova--Zhang and I'll mention some results of other people.