On cosets in Coxeter groups. (Q2892197)

For a Coxeter group \(W\) and a subset \(X\) of \(W\), let NEWLINE\[NEWLINEN(X)=\{\alpha\in\Phi^+\mid w\cdot\alpha\in\Phi^-\text{ for some }w\in X\},NEWLINE\]NEWLINE where \(\Phi^+\) and \(\Phi^-\) are, respectively, the positive and negative roots of the root system \(\Phi\) of \(W\). The Coxeter length of \(X\), \(l(X)\), is defined to be the cardinality of \(N(X)\).NEWLINENEWLINE Suppose that \(X\) is a subgroup of \(W\). For right cosets \(Xg\) and \(Xh\), write \(Xg\sim Xh\) whenever \(Xgt=Xh\) for some reflection \(t\in W\) and \(l(Xg)=l(Xh)\). Then \(\sim\) generates an equivalence relation on the right cosets of \(X\) in \(W\). Let \(\mathfrak X\) be the set of its equivalence classes.NEWLINENEWLINE Let \(\mathbf{x,x'}\in\mathfrak X\). Write \(\mathbf x\rightsquigarrow\mathbf x'\) if there is a right coset \(Xg\in\mathbf x\) and a right coset \(Xh\in\mathbf x'\) such that \(Xgt=Xh\) for some reflection \(t\) in \(W\) and \(l(Xg)\leq l(Xh)\). The partial order \(\preceq\) on \(\mathfrak X\) is defined by \(\mathbf x\preceq\mathbf x'\) if and only if there exist \(\mathbf x_1,\dots,\mathbf x_m\in\mathfrak X\) such that \(\mathbf x\rightsquigarrow\mathbf x_1\rightsquigarrow\cdots\rightsquigarrow\mathbf x_m\rightsquigarrow\mathbf x'\). Then \(\mathfrak X\) is called the \(X\)-poset of \(W\).NEWLINENEWLINE The authors obtain a number of results on \(X\)-posets, e.g., the \(X\)-posets represent a generalization of the Bruhat order on right cosets of standard parabolic subgroups, and also that certain \(X\)-posets are ranked and graded.