A note on element centralizers in finite coxeter groups
Röver, Claas E.
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Konvalinka, Matjaž; Pfeiffer, Götz; Röver, Claas E. (2011). A note on element centralizers in finite coxeter groups. Journal of Group Theory 14 (5), 727-745
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 epsilon 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). Then we use this result to give a new short proof of Solomon's Character Formula and discuss its connection to MacMahon's master theorem.