Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 042, 49 pages      arXiv:1409.7855      https://doi.org/10.3842/SIGMA.2015.042

Simplex and Polygon Equations

Aristophanes Dimakis a and Folkert Müller-Hoissen b
a) 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

Received October 23, 2014, in final form May 26, 2015; Published online June 05, 2015

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.

Key words: higher Bruhat order; higher Tamari order; pentagon equation; simplex equation.

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