On finite factors of centralizers of parabolic subgroups in Coxeter groups. (Q1936049)

Let \(W\) be a Coxeter group with \(S\) its Coxeter generator set. For any \(I\subseteq S\), denote by \(W_I\) the standard parabolic subgroup of \(W\) generated by \(I\). The structure of the centralizer \(Z_W(W_I)\) of \(W_I\) in \(W\) has been well studied. Consider the special case where \(I\) consists of a single reflection \(r\). We have a decomposition \(Z_W(r)=\langle r\rangle\times(W^{\perp r}\rtimes Y_r)\), where \(W^{\perp r}\) denotes the subgroup generated by all the reflections except \(r\) itself that commute with \(r\), and \(Y_r\) is a subgroup isomorphic to the fundamental group of a certain graph associated to \((W,S)\). It is known that \(W^{\perp r}\) is a Coxeter group. The present paper asserts that if \(s_\gamma\) is a Coxeter generator of \(W^{\perp r}\) belonging to a finite irreducible component of \(W^{\perp r}\), then \(s_\gamma\) commutes with every element of \(Y_r\). The main theorem of the paper further generalizes this result to the case of centralizers \(Z_W(W_I)\) of \(W_I\) for any \(I\subseteq S\) with the property that \(W_I\) has no irreducible component of type \(A_n\) with \(2\leqslant n<\infty\). The results of the paper can be applied to deal with some isomorphism problems in Coxeter groups.