Simplex and Polygon Equations

TL;DR

This paper explores the structure of higher Bruhat orders and their decomposition into Tamari orders, introducing polygon equations as generalizations of the pentagon equation, and visualizing these through polyhedral deformations.

Contribution

It presents a novel decomposition of higher Bruhat orders into Tamari and dual Tamari orders, and introduces polygon equations generalizing the pentagon equation.

Findings

Higher Bruhat orders decompose into Tamari, dual Tamari, and mixed orders.

Polygon equations generalize the pentagon equation.

Decomposition reduces N-simplex equations to polygon and dual equations.

Abstract

It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order." We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of "polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the $N$-simplex equation to the $(N+1)$-gon equation, its dual, and a compatibility equation.