Multislant matrices and Jacobi--Trudi determinants over finite fields

TL;DR

This paper derives formulas for the probability that determinants of certain structured matrices over finite fields vanish, generalizing previous results and exploring distribution properties of determinants in combinatorial matrix classes.

Contribution

It provides a new formula for the vanishing probability of determinants of multislant matrices with arithmetic progression partitions, extending prior conjectures.

Findings

Derived formulas for multislant matrices with arithmetic progression partitions.

Proved the determinant vanishing probability for a broad class of Toeplitz block matrices.

Showed that skew Jacobi-Trudi matrices are equidistributed over finite fields for ribbon partitions.

Abstract

The problem of counting the $\mathbb{F}_q$-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. (2018), which is similar to work of Kontsevich, Elkies, and Haglund. 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.