A note on element centralizers in finite Coxeter groups

Konvalinka, Matjaž
Pfeiffer, Götz
Röver, Claas E.
Konvalinka, Matjaž, Pfeiffer, Götz, & Röver Claas, E. (2011). A note on element centralizers in finite Coxeter groups, Journal of Group Theory (Vol. 14, pp. 727), DOI: 10.1515/jgt.2010.074
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.
De Gruyter
Publisher DOI
Attribution-NonCommercial-NoDerivs 3.0 Ireland