Simplex and Polygon Equations

2014/09/30 by Aristophanes Dimakis, Folkert Müller-Hoissen · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic structures and combinatorial models #Combinatorics #Computer science #Mathematics #Matrix Theory and Algorithms #Nonlinear Waves and Solitons #Polygon (computer graphics) #Pure mathematics #Simplex #math-ph #math.MP #math.QA #msc:06A06 #msc:06A07 #msc:52Bxx #msc:82B23 #nlin.SI

paper · pdf · doi:10.3842/sigma.2015.042

published as SIGMA 11 (2015), 042, pp. 49

arxiv created 2015/06/05 · openalex publication_date 2015/06/05 · arxiv updated 2015/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• The geometry of oriented cubes
• Special 2-cocycles and 3--3 Pachner move relations in Grassmann algebra
• Yang-Baxter maps: dynamical point of view
• On a two dimensional system associated with the complex of the solutions of the Tetrahedron equation
• Quantization of Lie bialgebras, V
• Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation
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• On Function Sum Theorems Connected with the Series ∑n=1∞xnn2

Cited by

• From NLS type matrix refactorisation problems to set-theoretical solutions of the 2- and 3-simplex equations
• Multicomponent pentagon maps

Related

• Constructing solutions of simplex equations from polygon equations
• The geometry of oriented cubes
• Pentagon and hexagon equations following Furusho
• Equations over Polyhedral Semirings
• Set-theoretical solutions of simplex equations