On finite factors of centralizers of parabolic subgroups in Coxeter groups

Details On finite factors of centralizers of parabolic subgroups in Coxeter groups On finite factors of centralizers of parabolic subgroups in Coxeter groups

2012-01-08

arxiv.org/abs/1201.1610v1

Mathematics

Group Theory

44 pages, 4 figures, 19 tables, separated from Section 7 of arXiv:math/0501061v4 [math.GR] with improvements

Scientific paper

It has been known that the centralizer $Z_W(W_I)$ of a parabolic subgroup $W_I$ of a Coxeter group $W$ is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $\mathcal{Y}$. In this paper, we study the structure of $Z_W(W_I)$ further and show that, if $I$ has no irreducible components of type $A_n$ with $2 \leq n < \infty$, then every element of finite irreducible components of the inner factor is fixed by a natural action of the fundamental group of $\mathcal{Y}$. This property has an application to the isomorphism problem in Coxeter groups.

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