On Some Properties of Algebraic and Combinatorial Invariants Induced by Brauer Configurations and Their Applications to the Solutions of the Yang–Baxter Equation

Green and Schroll introduced Brauer configurations to construct Brauer graph algebras and their generalizations, named Brauer configuration algebras, to investigate algebras of tame and wild representation types. It is worth pointing out that giving closed formulas for algebraic and combinatorial invariants (such as their dimensions, the dimensions of their centers, or degree sequences of their induced covering graphs) associated with significant families of Brauer configurations is, in general, a hard problem. The analysis of such algebraic and combinatorial invariants is said to be an extended Brauer analysis of the data defining the configurations. Since finite graphs are examples of Brauer configurations, this paper gives formulas for the dimensions of their induced Brauer configuration algebras and corresponding centers. Brauer configurations also induce some simple graphs called covering graphs. To date, it is not clear which properties a given graph must satisfy to be isomorphic to its induced covering graph. This paper fills this gap by establishing that the so-called hair graphs satisfy this condition. This approach allows us to use properties of morphisms in the graph category to provide set-theoretical solutions of the Yang–Baxter equation. Along these lines, we also note that giving a complete classification of the solutions to the Yang–Baxter equation remains an open problem.