Maximal length parabolic subgroups in finite Coxeter groups. (Q2883287)

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 \(N(X)\). Let \(X\) be a subgroup of a finite Coxeter group \(W\) and \(X^W\) be the set of \(W\)-conjugates of \(X\). Let \(f_X^+\) be the number of positive roots fixed by \(X\).NEWLINENEWLINE The following two theorems are among the main results obtained by the authors:NEWLINENEWLINE 1. For \(X\) a parabolic subgroup of a finite Coxeter group \(W\), NEWLINE\[NEWLINE\mu(X^W)= \max\{l(Y):Y\in X^W\}=|\Phi^+|-f_X^+.NEWLINE\]NEWLINE 2. Suppose that \(W\) is a Coxeter group of type \(A_n\), \(B_n\) or \(D_n\), that \(X\) is a standard parabolic subgroup of \(W\) and \(Y\) is a \(W\)-conjugate of \(X\). Then \(Y\) has maximal length in the \(W\)-conjugacy class of \(X\) if and only if the \(Y\)-orbits of \(\Omega\) form an intertwined partition of \(\Omega\). Here \(\Omega\) is a set of size \(n+1\) if \(W\) is of type \(A_n\) and of size \(2n\) if \(W\) is of type \(B_n\) or \(D_n\) and \(W\) acts as a permutation group on \(\Omega\).