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arXiv · 1611.00216

Jacobi-Trudi determinants over finite fields

Ben Anzis·Shuli Chen·Yibo Gao·Jesse Kim·Zhaoqi Li·Rebecca Patrias
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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 $\mathbb{F}_q$.

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Ben Anzis, Shuli Chen, Yibo Gao, Jesse Kim, Zhaoqi Li, Rebecca Patrias. 2016-11-01. Jacobi-Trudi determinants over finite fields. https://doi.org/10.1007/s00026-018-0399-8

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