ALGEBRAIC COMBINATORICS

Counting on the variety of modules over the quantum plane
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 583-592.

Let $\zeta$ be a fixed nonzero element in a finite field ${𝔽}_{q}$ with $q$ elements. In this article, we count the number of pairs $\left(A,B\right)$ of $n×n$ matrices over ${𝔽}_{q}$ satisfying $AB=\zeta BA$ 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 $XY=\zeta YX$, whose geometry was described by Chen and Lu.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.230
Classification: 10X99,  14A12,  11L05
Keywords: quantum plane, matrix equations, counting over finite fields, generating functions
Huang, Yifeng 1

1 University of Michigan Dept. of Mathematics 530 Church St Ann Arbor MI 48109 (USA)
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Huang, Yifeng. Counting on the variety of modules over the quantum plane. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 583-592. doi : 10.5802/alco.230. https://alco.centre-mersenne.org/articles/10.5802/alco.230/

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