Symmetric Group Action of the Birational $R$-matrix

TL;DR

This paper explores the symmetric group action induced by the birational R-matrix, providing explicit formulas and combinatorial interpretations for permutation actions beyond simple transpositions.

Contribution

It introduces explicit formulas and combinatorial interpretations for the action of various permutations in the symmetric group via the birational R-matrix.

Findings

Formulas for permutation actions beyond simple transpositions

Combinatorial interpretations of functions involved

Extension of known results to additional cases

Abstract

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 precisely the formula corresponding to the action of the simple transposition $s_i$, 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.