On the Structure of Set-Theoretic Polygon Equations

TL;DR

This paper explores the structure and reductions of polygon equations, generalizations of the pentagon equation, revealing relations between solutions and their connections to higher Bruhat and Tamari orders.

Contribution

It introduces reductions of polygon equations and their duals, explicitly analyzing their structure up to the octagon equation, and links these to higher combinatorial orders.

Findings

Reductions of polygon equations relate solutions of neighboring equations.

Explicit analysis of polygon equations up to the octagon.

Connections established between polygon equations and higher combinatorial orders.

Abstract

Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the $(N-1)$-simplex equation can be regarded as a realization of the higher Bruhat order $B(N,N-2)$, the $N$-gon equation is a realization of the higher Tamari order $T(N,N-2)$. The latter and its dual $\tilde T(N,N-2)$, associated with which is the dual $N$-gon equation, have been shown to arise as suborders of $B(N,N-2)$ via a ''three-color decomposition''. There are two different reductions of $T(N,N-2)$ and $\tilde T(N,N-2)$, to ${T(N-1,N-3)}$, respectively $\tilde T(N-1,N-3)$. In this work, we explore the corresponding reductions of (dual) polygon equations, which lead to relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation.