Combinatorics Seminar-Rook theory of the finite general linear group

Speaker: Joel Brewster Lewis, GWU
Date and time: Wednesday, November 15, 4-5pm
Place: Rome 771
Title: Rook theory of the finite general linear group
Abstract: The combinatorial field of rook theory considers the following questions: given a subset B (called a board) of the discrete n-by-n square, how many ways are there to place r rooks on the board so that no two lie in the same row or column?  And how many of the full placements of n rooks intersect B in exactly r squares?  These numbers, respectively called the rook number and hit number, satisfy a variety of pleasant identities and dualities.

Beginning with work of Garsia and Remmel in the 1980s, combinatorialists have considered q-analogues of these problems, in the following sense: associate to each rook placement P a statistic stat(P), and compute the sum of q^(stat(P)) over all rook placements, where q is a formal variable.  The result is a polynomial in q whose coefficients count rook placements according to stat, and whose value at q = 1 is the rook number.  Haglund showed that for sufficiently nice boards (the so-called Ferrers diagrams), the Garsia--Remmel q-rook number can also be obtained by counting matrices of a given rank whose support is in the board B over a finite field.  In this talk, I'll describe recent work with Morales in which we further investigate this finite field rook theory, including a definition of a q-analogue of the hit numbers and a number of intriguing questions related to positivity of certain formulas.