对合词II：辫状关系和原子结构
Involution words II: braid relations and atomic structures
摘要：
对合字是Coxeter群中扭曲对合的约化字的变体。他们自然出现在研究的布鲁哈特秩序，某些岩堀-赫克代数模块，并关闭轨道的旗帜品种。具体地说，对于具有自同构$*$的Coxeter群$W$中的任意扭曲对合$x$、$y$，我们关联一组对合字$\hat{\mathcal{R}}_*(x,y)$。这个集合是一组群元素$\mathcal{A}_*(x,y)$的约化字的不交并，我们称之为相对于$x$的$y$的原子。这些原子又包含在一个更大的集合$\mathcal{B}_*(x,y) \subset W$中，这个集合具有类似的定义，我们把它的元素称为赫克原子。我们的主要结果涉及集合$\hat{\mathcal{R}}_*(x,y)$和$\mathcal{A}_*(x,y) \subset \mathcal{B}_*(x,y)$的一些有趣的性质。对于有限Coxeter群，证明了$\mathcal{A}_*(1,y)$恰好由最小长度元$w \in W$组成，使得$w^* y \leq w$为Bruhat阶，并猜想了任意Coxeter群的一个更一般的性质.在$A$型中，我们描述了一组简单的条件，描述了所有对合$x,y \in S_n$的集合$\mathcal{A}_*(x,y)$的特征，给出了Can，Joyce和Wyser最近三个定理的共同推广。我们证明了对称群中固定对合的原子（相对于$x=1$）自然地形成一个分次偏序集，而Hecke原子在Cassaigne，Espie等人研究的“中国关系”下令人惊讶地形成一个等价类。这些事实使我们能够恢复胡和张的一个最近的定理，该定理描述了一组“辫子关系”，它跨越任何自逆置换的对合字。我们证明了这一结果的推广给出了一个类似的松本定理对合字在任意Coxeter群。
英文摘要：
Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori-Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted involutions $x$, $y$ in a Coxeter group $W$ with automorphism $*$, we associate a set of involution words $\hat{\mathcal{R}}_*(x,y)$. This set is the disjoint union of the reduced words of a set of group elements $\mathcal{A}_*(x,y)$, which we call the atoms of $y$ relative to $x$. The atoms, in turn, are contained in a larger set $\mathcal{B}_*(x,y) \subset W$ with a similar definition, whose elements we refer to as Hecke atoms. Our main results concern some interesting properties of the sets $\hat{\mathcal{R}}_*(x,y)$ and $\mathcal{A}_*(x,y) \subset \mathcal{B}_*(x,y)$. For finite Coxeter groups we prove that $\mathcal{A}_*(1,y)$ consists of exactly the minimal-length elements $w \in W$ such that $w^* y \leq w$ in Bruhat order, and conjecture a more general property for arbitrary Coxeter groups. In type $A$, we describe a simple set of conditions characterizing the sets $\mathcal{A}_*(x,y)$ for all involutions $x,y \in S_n$, giving a common generalization of three recent theorems of Can, Joyce, and Wyser. We show that the atoms of a fixed involution in the symmetric group (relative to $x=1$) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the "Chinese relation" studied by Cassaigne, Espie, et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of "braid relations" spanning the involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto's theorem for involution words in arbitrary Coxeter groups.
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