Multislant matrices and Jacobi–Trudi determinants over finite fields

Abstract

The problem of counting the Fq-valued points of a variety has been well-studied from algebro-geometric, topological, and combinatorial perspectives. We explore a combinatorially flavored version of this problem studied by Anzis et al. [1], which is similar to work of Kontsevich [7], Elkies [4], and Haglund [5].Anzis et al. considered the question: what is the probability that the determinant of a Jacobi-Trudi matrix vanishes if the variables are chosen uniformly at random from a finite field? They gave a formula for various partitions such as hooks, staircases, and rectangles. We give a formula for partitions whose parts form an arithmetic progression, verifying and generalizing one of their conjectures. More generally, we compute the probability of the determinant vanishing for a class of matrices (“multislant matrices”) made of Toeplitz blocks with certain properties.We furthermore show that the determinant of a skew Jacobi-Trudi matrix is equidistributed across the finite field if the skew partition is a ribbon.