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Simplex and polygon equations. (English) Zbl 1338.06001

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.

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
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