On the Structure of Set-Theoretic Polygon Equations

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.

References in corpus (7)

• Simplex and Polygon Equations
• Matrix factorizations and pentagon maps
• Idempotent set-theoretical solutions of the pentagon equation
• Set-theoretical solutions of the pentagon equation on Clifford semigroups
• On Pentagon Equation, Tetrahedron Equation, Evolution and Integrals of Motion
• Quantum invariants of closed framed $3$-manifolds based on ideal triangulations
• Algebraic formulas whose structure imitates Pachner moves and new types of acyclic complexes

Cited by in corpus (1)

• Set-theoretical solutions to the pentagon equation: a survey