Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 051, 30 pages      arXiv:2305.17974      https://doi.org/10.3842/SIGMA.2024.051

On the Structure of Set-Theoretic Polygon Equations

Folkert Müller-Hoissen
Institut für Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

Received December 29, 2023, in final form May 29, 2024; Published online June 11, 2024

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.

Key words: polygon equations; simplex equations, cocycle equations; pentagon equation; set-theoretic solutions; higher Bruhat orders; higher Tamari orders.

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