Quotients of twisted Bruhat orders
The present paper is a continuation of the author's preceding paper [Contemp. Math. 139, 141-165 (1992; Zbl 0833.20048)] on the study of a twisted Bruhat order \(\leq_A\) on a Coxeter group \(W\), where \(A\) is some initial section of reflections of \(W\). Let \(W\) be an indecomposable affine Weyl group represented geometrically on a euclidean space \(V\) with inner product \((\mid)\) and a choice of positive root system \(\Phi^+\) of \(W\) spanning \(V\). For a regular element \(\mu\in V\), let \(A=\{s_\alpha \mid \alpha \in \Phi^+,\;(\alpha|\mu)>0\}\). Then the author shows that \(W\) in the order \(\leq_A\) is order isomorphic to either Bruhat order, reverse Bruhat order, or the alcove order defined by \textit{G. Lusztig} [Adv. Math. 37, 121-164 (1980; Zbl 0448.20039)]. On the other hand, the author considers the quotients of a twisted Bruhat order of a Coxeter group \(W\). By the use of restrictions of ``reflection orders'' to their initial sections as substitutes for reduced expressions, the author extends some results of \textit{A. Björner} and \textit{M. Wachs} [Trans. Am. Math. Soc. 308, 1-37 (1988; Zbl 0659.05007)] on generalized quotients of Bruhat order of \(W\), to those on quotients of a twisted Bruhat order of \(W\).
scientific article; zbMATH DE number 554154