This article is cited in 24 papers

Simplex and Polygon Equations

Aristophanes Dimakis^a, Folkert Müller-Hoissen<sup>b

a</sup> Department of Financial and Management Engineering, University of the Aegean, 82100 Chios, Greece
^b Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

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.

Keywords: higher Bruhat order; higher Tamari order; pentagon equation; simplex equation.

MSC: 06A06; 06A07; 52Bxx; 82B23

Received: October 23, 2014; in final form May 26, 2015; Published online June 5, 2015

Language: English

DOI: 10.3842/SIGMA.2015.042

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ArXiv: 1409.7855