## Tsukuba Journal of Mathematics

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

Koji Nuida

#### Abstract

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

#### Article information

Source
Tsukuba J. Math., Volume 36, Number 2 (2013), 235-294.

Dates
First available in Project Euclid: 21 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1358777001

Digital Object Identifier
doi:10.21099/tkbjm/1358777001

Mathematical Reviews number (MathSciNet)
MR3058241

Zentralblatt MATH identifier
1276.20045

Subjects
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 20E34: General structure theorems

#### Citation

Nuida, Koji. On finite factors of centralizers of parabolic subgroups in Coxeter groups. Tsukuba J. Math. 36 (2013), no. 2, 235--294. doi:10.21099/tkbjm/1358777001. https://projecteuclid.org/euclid.tkbjm/1358777001