Set-theoretic solutions of the Pentagon Equation

Abstract

A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s:S^2\to S^2$ such that $s_{23}{s}_{13}{s}_{12}=s_{12}{s}_{23}$, where $s_{12}=s\times \mathrm{id}$, $s_{23}=\mathrm{id}\times s$ and $s_{13}=(\tau \times \mathrm{id})(\mathrm{id}\times s)(\tau \times \mathrm{id})$ are mappings from $S^3$ to itself and $\tau :S^2\to S^2$ is the flip map, i.e., $\tau (x,y)=(y,x)$. We give a description of all involutive solutions, i.e., $s^2=\mathrm{id}$. It is shown that such solutions are determined by a factorization of $S$ as direct product $X\times A\times G$ and a map $\sigma :A\to \mathrm{Sym}(X)$, where $X$ is a non-empty set and $A,G$ are elementary abelian $2$-groups. Isomorphic solutions are determined by the cardinalities of $A$, $G$ and $X$, i.e., the map $\sigma$ is irrelevant. In particular, if $S$ is finite of cardinality $2^n(2m+1)$ for some $n,m\ge 0$ then, on $S$, there are precisely $\left(\genfrac{}{}{0ex}{}{n+2}{2}\right)$ non-isomorphic solutions of the Pentagon Equation.

accepted for publication in Communication in Mathematical Physics. 16 pages