论文标题
对称的小组动作，$ r $ -matrix
Symmetric Group Action of the Birational $R$-matrix
论文作者
论文摘要
Birational $ r $ -Matrix是一种转换，它出现在几何晶体理论中，对环组的总阳性研究以及离散的动态系统。这个$ r $ -matrix引起了对称组$ s_m $在$ m $ $ $ $ tuple的动作。虽然Birational $ r $ -Matrix恰恰是与简单转置$ s_i $的动作相对应的公式，但通常不知道针对其他排列行动的显式公式。 Lam和Pylyavskyy研究了一种特殊情况，因为它与晶体的能量功能有关。在本文中，我们将讨论其他几个其他情况，包括换位，并为我们工作中出现的功能提供组合解释。
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.
