论文标题
在简化单词集的图表上
On Graphs of Sets of Reduced Words
论文作者
论文摘要
有限对称组中的任何置换都可以写成简单换位的产物$ s_i =（i〜i+1）$。对于固定的排列，$σ\ in \ mathfrak {s} _n $最小长度的产物称为减少分解或简化单词，并且所有此类简化单词的收集均表示为$ \ Mathcal {r}（r}（σ）$。任何简化的$σ$的单词都可以通过一系列换向动作或长辫子移动来转换为其他任何单词。这些集合中感兴趣的领域是仅使用辫子或仅换向关系来定义的一致性类别。可以将$ \ MATHCAL {r}（σ）$绘制为图形，$ g（σ）$，其中顶点是简化的单词，边缘表示在单词之间的换向或辫子移动。本文介绍了$ g（σ）$中的子图结构的新作品，以及新的公式，以计算$ g（σ）$中的编织边缘和换向边缘的数量。我们还包括$ \ Mathcal {r}（σ）$中的编织和换向类数量的界限。
Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $σ\in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or reduced words, and the collection of all such reduced words is denoted $\mathcal{R}(σ)$. Any reduced word of $σ$ can be transformed into any other by a sequence of commutation moves or long braid moves. One area of interest in these sets are the congruence classes defined by using only braid or only commutation relations. The set $\mathcal{R}(σ)$ can be drawn as a graph, $G(σ)$, where the vertices are the reduced words, and the edges denote the presence of a commutation or braid move between the words. This paper presents new work on subgraph structures in $G(σ)$, as well as new formulas to count the number of braid edges and commutation edges in $G(σ)$. We also include work on bounds for the number of braid and commutation classes in $\mathcal{R}(σ)$.
