Counting on the variety of modules over the quantum plane

DOI10.5802/ALCO.230arXiv2110.15570MaRDI QIDQ6381641

Yifeng Huang

Publication date: 29 October 2021

Abstract: Let $zeta$ 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 $nimesn$ matrices over $mathbb{F}_q$ satisfying $AB=zetaBA$ 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 $xy=zetayx$, whose geometry was described by Chen and Lu.

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Quantum groups (quantized enveloping algebras) and related deformations (17B37) Matrices over special rings (quaternions, finite fields, etc.) (15B33) Commutativity of matrices (15A27) Special varieties (14M99) Representations of quivers and partially ordered sets (16G20) Simple, semisimple, reductive (super)algebras (17B20) Lie algebras of linear algebraic groups (17B45)

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