Geometry of Yang-Baxter matrix equations over finite fields

TL;DR

This paper investigates the geometric structure of solutions to the Yang-Baxter matrix equation over finite fields, explicitly characterizing solutions and deriving formulas for their counts.

Contribution

It introduces a computational ideal theory approach to explicitly describe all solutions and determine their cardinalities for the Yang-Baxter matrix equation over finite fields.

Findings

Explicit solutions to the matrix equation are provided.

Cardinality formulas for the solution varieties are derived.

The geometric structure of the solution set is characterized.

Abstract

Let $A$ be a $2\times 2$ matrix over a finite field and consider the Yang-Baxter matrix equation $XAX=AXA$ with respect to $A$. We use a method of computational ideal theory to explore the geometric structure of the affine variety of all solutions to this equation. In particular, we exhibit all solutions explicitly and determine cardinality formulas for these varieties.