论文标题
Coxeter组中的结构
Structure of conjugacy classes in Coxeter groups
论文作者
论文摘要
本文为在循环偏移方面描述任意Coxeter组中描述共轭类别的问题提供了明确的解决方案。 令$（W，S）$为Coxeter系统。 W $中元素$ w \的循环移动是S $中的某些简单反射$ s \ SWS $的$ W $的共轭，因此$ \ ell_s（sws）\ ell_s（sws）\ leq \ ell_s（w）$。 $ w $的循环偏移类是$ w $的元素集，可以通过一系列循环偏移从$ w $获得。给定一个子集$ k \ subseteq s $，使得$ w_k：= \ langle k \ rangle \ subseteq w $是有限的，我们也称为两个元素$ w，w $ $ k $ -k $ -conjugate，如果$ w，如果$ w，如果$ w，如果$ W_K $。 令$ \ mathcal o $为$ W $中的共轭类，让$ \ Mathcal o^{\ min} $成为$ \ Mathcal o $的最小长度元素的集合。然后，$ \ Mathcal o^{\ min} $是有限的许多环形移位类别$ C_1，\ dots，c_k $的不相交联盟。我们将与$ \ Mathcal O $相关的结构共轭图定义为具有顶点$ C_1，\ dots，c_k $的图形，并且在不同的顶点$ C_I，C_J $之间具有优势，如果它们包含c_i $中的代表$ u \ u \ in C_I $和$ v \ in C_j $ in c_j $，则$ v $ s $ k $ kj $ kj $ kjugate $ kjjugate $ kjugate $ kconjugate $ conjugate $ conjugate $ conjugate $ conjugate。 在本文中，我们明确计算了与$ W $中任何（可能扭曲的）共轭类相关的结构共轭图，并特别表明它已连接（即，$ W $的任何两个共轭元素只有一系列循环移位和$ k $ -conjugations差异）。在此过程中，我们获得了几个独立兴趣的结果，例如对W $中无限顺序元素$ w \的中心人的描述，以及$ w $的天然分解，作为“扭转部分”的产物和“直部”的产物，具有有用的属性。
This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let $(W,S)$ be a Coxeter system. A cyclic shift of an element $w\in W$ is a conjugate of $w$ of the form $sws$ for some simple reflection $s\in S$ such that $\ell_S(sws)\leq\ell_S(w)$. The cyclic shift class of $w$ is then the set of elements of $W$ that can be obtained from $w$ by a sequence of cyclic shifts. Given a subset $K\subseteq S$ such that $W_K:=\langle K\rangle\subseteq W$ is finite, we also call two elements $w,w'\in W$ $K$-conjugate if $w,w'$ normalise $W_K$ and $w'=w_0(K)ww_0(K)$, where $w_0(K)$ is the longest element of $W_K$. Let $\mathcal O$ be a conjugacy class in $W$, and let $\mathcal O^{\min}$ be the set of elements of minimal length in $\mathcal O$. Then $\mathcal O^{\min}$ is the disjoint union of finitely many cyclic shift classes $C_1,\dots,C_k$. We define the structural conjugation graph associated to $\mathcal O$ to be the graph with vertices $C_1,\dots,C_k$, and with an edge between distinct vertices $C_i,C_j$ if they contain representatives $u\in C_i$ and $v\in C_j$ such that $u,v$ are $K$-conjugate for some $K\subseteq S$. In this paper, we compute explicitely the structural conjugation graph associated to any (possibly twisted) conjugacy class in $W$, and show in particular that it is connected (that is, any two conjugate elements of $W$ differ only by a sequence of cyclic shifts and $K$-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element $w\in W$, as well as the existence of natural decompositions of $w$ as a product of a "torsion part" and of a "straight part", with useful properties.
