Jacobi-Trudi determinants over finite fields

TL;DR

This paper investigates the probability that Schur functions over symmetric functions vanish when mapped to finite fields, providing bounds, classifications, and independence results for various shapes.

Contribution

It offers a complete classification of shapes with probability $1/q$ and analyzes independence and non-vanishing probabilities of Schur functions over finite fields.

Findings

Probability of vanishing is always at least 1/q

Asymptotic probability of vanishing approaches 1/q

Classification of shapes achieving probability 1/q

Abstract

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_\lambda$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.