Dimakis, Aristophanes; Müller-Hoissen, Folkert Simplex and polygon equations. (English) Zbl 1338.06001 SIGMA, Symmetry Integrability Geom. Methods Appl. 11, Paper 042, 49 p. (2015). Summary: 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. Cited in 18 Documents MSC: 06A06 Partial orders, general 06A07 Combinatorics of partially ordered sets 16T25 Yang-Baxter equations 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 82B23 Exactly solvable models; Bethe ansatz Keywords:higher Bruhat orders; higher Tamari orders; Yang-Baxter equation; pentagon equation; simplex equations PDFBibTeX XMLCite \textit{A. Dimakis} and \textit{F. Müller-Hoissen}, SIGMA, Symmetry Integrability Geom. Methods Appl. 11, Paper 042, 49 p. (2015; Zbl 1338.06001) Full Text: DOI arXiv EMIS