CL-shellability of ordered structure of reflection systems (Q1277021)

Let \((X,E)\) be an undirected finite graph with a vertex set \(X\) and an edge set \(E\) and let \(\varphi:X\to \{n\in \mathbb{N}\mid n\geq 2\}\), \(\Psi:E\to \{n\in \mathbb{N}\mid n\geq 3\}\cup \{\infty\}\) where \(\varphi(x)= \Psi(y)=2\) whenever \(\{x,y\}\in E\) and \(\Psi(x,y)\neq \infty\). The pair \((G,X)\) of a group \(G\) and its system of generators \(X\) is called a reflection system if \(G\) satisfies the following relations (1) \(x^{\varphi(x)}=1\) for \(x\in X\); (2) \(xy=yx\) if \(\{x,y\}\not\in E\); (3) \((xy)\Psi(\{x,y\})=1\) if \(\{x,y\}\in E\) and \(\Psi(\{x,y\})< \infty\). Note that Coxeter systems are reflection systems with \(\varphi(x)=2\) for all \(x\in X\). Reflection systems have associated Coxeter (like) diagrams and they permit Bruhat (like) orders which reduce to Bruhat orders for Coxeter systems/groups. Using a carefully presented clever set of modifications and some new tricks the author is able to reestablish the Björner-Wachs result on CL-shellability of Bruhat orders to this larger class of closed intervals of the generalized Bruhat order of reflection systems. For students interested in an introduction to this theory the preliminary material provides a good starting point permitting expansion into the past of the subject in a surprisingly smooth manner.