Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups (Q6662816)

Let \((W,S)\) be a Coxeter system, \(J\) a subset of \(S\), and defines \(W_{J}\) the subgroup of \(W\) generated by \(J\). The pair \((W_{J},J)\) is again a Coxeter system, with Coxeter-Dynkin diagram obtained from the diagram of \(W\) by removing the vertices corresponding to generators in \(S\setminus J\). In this situation (1) there is a unique element \(x^{J}\in W_{J}\) which has minimal length among all elements in \(xW_{J}\); (2) for all \(y \in W_{J}\), one has \(x^{J} \leq y\), where \(\leq\) is the strong Bruhat order on \(W\).\N\NThe paper under review is dedicated to the study restriction of the strong Bruhat order on an arbitrary Coxeter group \(W\) to cosets \(xW^{\theta}_{L}\), where \(x \in W\) and \(W_{L}^{\theta}\) is the subgroup of fixed points of an automorphism \(\theta\) of order at most two of a standard parabolic subgroup \(W_{L}\) of \(W\). If \(\theta \not = \mathrm{id}\), then there is in general more than one element of minimal length in a given coset and the authors explain how to relate elements of minimal length. They also show that elements of minimal length in cosets are exactly those elements which are minimal for the restriction of the Bruhat order.