Symmetric group action of the birational $R$-matrix

The birational $R$-matrix is a transformation that appears in the theory of geometric crystals, the study of total positivity in loop groups, and discrete dynamical systems. This $R$-matrix gives rise to an action of the symmetric group $S_m$ on an $m$-tuple of vectors. While the birational $R$-matrix is defined by the action of the simple transpositions $s_i$, explicit formulas for the action of other permutations are generally not known. One particular case was studied by Lam and Pylyavskyy as it relates to energy functions of crystals. In this paper, we will discuss formulas for several additional cases, including transpositions, and provide combinatorial interpretations for the functions that appear in our work.

Keywords

birational $R$-matrix, networks on a cylinder

2010 Mathematics Subject Classification

Primary 05E18. Secondary 15A23, 22E67, 81R50.

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The authors’ research was partially conducted at the 2020 University of Minnesota Twin Cities REU, which was supported by NSF RTG grant DMS-1745638.

Received 16 July 2021

Published 28 December 2022