Simplex and Polygon 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.

References in corpus (2)

• Yang-Baxter maps: dynamical point of view
• The geometry of oriented cubes

Cited by in corpus (15)

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• Grassmannian-parameterized solutions to direct-sum polygon and simplex equations
• Matrix factorizations and pentagon maps
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Idempotent set-theoretical solutions of the pentagon equation
• Birational solutions to the set-theoretical 4-simplex equation
• On the Structure of Set-Theoretic Polygon Equations
• Set-theoretical solutions of the pentagon equation on Clifford semigroups
• Odd-gon relations and their cohomology
• Matrix KP: tropical limit, Yang-Baxter and pentagon maps
• Set-theoretical solutions to the pentagon equation: a survey
• Heptagon relations parameterized by simplicial 3-cocycles
• Heptagon relations from a simplicial 3-cocycle, and their cohomology