From indecomposable solutions to the Yang--Baxter equation to irreducible representations

What does the indecomposability of a set-theoretical solution of the Yang–Baxter equation have to do with representation theory? In joint work with C. Dietzel and E. Feingesicht, we explore this connection for involutive, non-degenerate solutions by studying monomial representations of their associated structure groups and their Coxeter-like quotients. Using the brace structure of these groups, we show that, in most cases, the irreducibility of these representations is equivalent to the indecomposability of the solution, while in the remaining cases we obtain sufficient conditions for irreducibility. We also obtain explicit descriptions of the representations as induced representations from one-dimensional characters of point stabilisers.

Sep 17, 2026