双有理$R$ -矩阵的对称群作用
Symmetric Group Action of the Birational $R$-matrix
摘要：
双有理$R$ -矩阵是一种变换，出现在几何晶体理论、环群的全正性研究和离散动力系统中。这个$R$ -矩阵引起对称群$S_m$对向量的$m$ -元组的作用.虽然双有理$R$ -矩阵精确地是对应于简单转置$s_i$的作用的公式，但是用于其他置换的作用的显式公式通常是未知的。Lam和Pylyavskyy研究了一种特殊情况，因为它与晶体的能量函数有关。在本文中，我们将讨论几个额外的情况下，包括换位公式，并提供组合解释的功能，出现在我们的工作。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
