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arXiv · 1409.7855

Simplex and Polygon Equations

Aristophanes Dimakis·Folkert Müller-Hoissen
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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.

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Aristophanes Dimakis, Folkert Müller-Hoissen. 2014-09-27. Simplex and Polygon Equations. https://doi.org/10.3842/sigma.2015.042

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