TY - JOUR

T1 - Locally finite continuations and Coxeter groups of infinite ranks

AU - Mühlherr, Bernhard

AU - Nuida, Koji

N1 - Funding Information:
A large portion of the present work was done when the second author was with National Institute of Advanced Industrial Science and Technology. A part of this work was supported by JST PRESTO Grant Number JPMJPR14E8 , JST CREST Grant Number JPMJCR14D6 , and JSPS KAKENHI Grant Number JP19H01804 to the second author. The discussion for one of the main parts of this work was undertaken at Tambara Institute of Mathematical Sciences, The University of Tokyo. The authors would like to thank Itaru Terada for kindly organizing our discussion in Tambara and giving helpful comments.
Funding Information:
A large portion of the present work was done when the second author was with National Institute of Advanced Industrial Science and Technology. A part of this work was supported by JST PRESTO Grant Number JPMJPR14E8, JST CREST Grant Number JPMJCR14D6, and JSPS KAKENHI Grant Number JP19H01804 to the second author. The discussion for one of the main parts of this work was undertaken at Tambara Institute of Mathematical Sciences, The University of Tokyo. The authors would like to thank Itaru Terada for kindly organizing our discussion in Tambara and giving helpful comments.
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2021/1

Y1 - 2021/1

N2 - An involution r in a Coxeter group W is called an intrinsic reflection of W if r∈SW for each Coxeter generating set S of W. In recent joint work with R.B. Howlett [13] we determined all intrinsic reflections in finitely generated Coxeter groups. In the present paper we extend this result to the infinite rank case. An important tool in [13] is the notion of the finite continuation of an involution that is only meaningful for finitely generated Coxeter groups. Here we introduce the locally finite continuation for any subset of an arbitrary group which enables us to deal with Coxeter groups of infinite rank. We apply our result to show that certain classes of Coxeter groups are reflection independent and we investigate rigidity of 2-spherical Coxeter systems of arbitrary ranks.

AB - An involution r in a Coxeter group W is called an intrinsic reflection of W if r∈SW for each Coxeter generating set S of W. In recent joint work with R.B. Howlett [13] we determined all intrinsic reflections in finitely generated Coxeter groups. In the present paper we extend this result to the infinite rank case. An important tool in [13] is the notion of the finite continuation of an involution that is only meaningful for finitely generated Coxeter groups. Here we introduce the locally finite continuation for any subset of an arbitrary group which enables us to deal with Coxeter groups of infinite rank. We apply our result to show that certain classes of Coxeter groups are reflection independent and we investigate rigidity of 2-spherical Coxeter systems of arbitrary ranks.

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U2 - 10.1016/j.jpaa.2020.106464

DO - 10.1016/j.jpaa.2020.106464

M3 - Article

AN - SCOPUS:85087693194

VL - 225

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 1

M1 - 106464

ER -