Centralizers of maps as automorphism groups of Yang--Baxter solutions

Abstract

For a partition $X = Y \sqcup Z$ of a set $X$ and an arbitrary map $g \colon Y \to {\rm Hom}_{\rm Set} (Z, \, Z)$, we construct a solution $R_g \colon X \times X \to X \times X$ of the braid equation. We characterize intrinsically the solutions which are isomorphic to some $R_g$ and we give a necessary and sufficient condition for two solutions $R_g$ and $R_{g'}$ to be isomorphic. Consequently, the number of isomorphism classes of the solutions $R_g$ on a set with $n$ elements is $1 + \sum_{k = 1}^{n-1} \widetilde{o} (n - k, \, k)$, where $\widetilde{o} (m, \, k)$ is the number of orbits of the symmetric group $S_m$, acting by conjugation, on the set of multisets with $k$ elements of maps $\{1, \ldots, m\} \to \{1, \ldots, m\}$. This construction is then used to obtain a solution of the braid equation whose automorphism group is exactly the centralizer ${\rm Aut} (f_1, \ldots, f_k)$ of a given family $f_1, \ldots, f_k \colon X \to X$ of maps, $k \geq 1$. For $k = 2$, combined with a result of Agore, Chirvasitu and the author, this gives another proof of the fact that any group is isomorphic to the automorphism group of a solution of the braid equation (equivalently, of the set-theoretic Yang--Baxter equation).