# A Note on Element Centralizers in Finite Coxeter Groups - Mathematics > Group Theory

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Abstract: The normalizer $N WW J$ of a standard parabolic subgroup $W J$ of a finiteCoxeter group $W$ splits over the parabolic subgroup with complement $N J$consisting of certain minimal length coset representatives of $W J$ in $W$. Inthis note we show that with the exception of a small number of cases arisingfrom a situation in Coxeter groups of type $D n$ the centralizer $C Ww$ ofan element $w \in W$ is in a similar way a semidirect product of thecentralizer of $w$ in a suitable small parabolic subgroup $W J$ with complementisomorphic to the normalizer complement $N J$.

Author: Matjaž Konvalinka, Götz Pfeiffer, Claas Röver

Source: https://arxiv.org/