A Note on Element Centralizers in Finite Coxeter Groups.

Matjaž Konvalinka, Götz Pfeiffer and Claas Röver

Abstract

The normalizer N_W(W_J) of a standard parabolic subgroup W_J of a finite Coxeter group W splits over the parabolic subgroup with complement N_J consisting of certain minimal length coset representatives of W_J in W. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type D_n) the centralizer C_W(w) of an element w of W is in a similar way a semidirect product of the centralizer of w in a suitable small parabolic subgroup W_J with complement isomorphic to the normalizer complement N_J.

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