Counting on the variety of modules over the quantum plane

2021/10/29 by Yifeng Huang, Huang, Yifeng
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2110.15570

openalex publication_date 2021/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ζ be a fixed nonzero element in a finite field \mathbb F_q with q elements. In this article, we count the number of pairs (A,B) of n× n matrices over \mathbb F_q satisfying AB=ζBA by giving a generating function. This generalizes a generating function of Feit and Fine that counts pairs of commuting matrices. Our result can be also viewed as the point count of the variety of modules over the quantum plane xy=ζyx, whose geometry was described by Chen and Lu.

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