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Multislant matrices and Jacobi-Trudi determinants over finite fields
Journal article   Open access   Peer reviewed

Multislant matrices and Jacobi-Trudi determinants over finite fields

FINITE FIELDS AND THEIR APPLICATIONS, v 91, 102262
Oct 2023
DOI: https://doi.org/10.1016/j.ffa.2023.102262
Featured in Collection :   UN Sustainable Development Goals @ Drexel
url
https://arxiv.org/pdf/2302.07239View
Submitted Open arXiv.org - Non-exclusive license to distribute

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 JacobiTrudi matrix is equidistributed across the finite field if the skew partition is a ribbon.& COPY; 2023 Elsevier Inc. All rights reserved.

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Web of Science research areas
Mathematics
Mathematics, Applied
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