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.

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Cited by in corpus (18)

• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Grassmannian-parameterized solutions to direct-sum polygon and simplex equations
• Matrix factorizations and pentagon maps
• Idempotent set-theoretical solutions of the pentagon equation
• Birational solutions to the set-theoretical 4-simplex equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Heptagon relation in a direct sum
• Pentagon identities arising in supersymmetric gauge theory computations
• 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
• $2^n-$rational maps