Algebraic & Geometric Topology

Matching theorems for systems of a finitely generated Coxeter group

Michael Mihalik, John G Ratcliffe, and Steven T Tschantz

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Abstract

We study the relationship between two sets S and S of Coxeter generators of a finitely generated Coxeter group W by proving a series of theorems that identify common features of S and S. We describe an algorithm for constructing from any set of Coxeter generators S of W a set of Coxeter generators R of maximum rank for W.

A subset C of S is called complete if any two elements of C generate a finite group. We prove that if S and S have maximum rank, then there is a bijection between the complete subsets of S and the complete subsets of S so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of (W,S) and (W,S) have the same multiset of entries.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 2 (2007), 919-956.

Dates
Received: 31 March 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796710

Digital Object Identifier
doi:10.2140/agt.2007.7.919

Mathematical Reviews number (MathSciNet)
MR2336245

Zentralblatt MATH identifier
1134.20045

Keywords
Coxeter groups

Citation

Mihalik, Michael; Ratcliffe, John G; Tschantz, Steven T. Matching theorems for systems of a finitely generated Coxeter group. Algebr. Geom. Topol. 7 (2007), no. 2, 919--956. doi:10.2140/agt.2007.7.919. https://projecteuclid.org/euclid.agt/1513796710


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References

  • N Bourbaki, Éléments de mathématique Fasc. XXXIV: Groupes et algèbres de Lie, Chapitres IV–VI, Actualités Scientifiques et Industrielles 1337, Hermann, Paris (1968)
  • N Brady, J P McCammond, B Mühlherr, W D Neumann, Rigidity of Coxeter groups and Artin groups, Geom. Dedicata 94 (2002) 91–109
  • B Brink, R B Howlett, Normalizers of parabolic subgroups in Coxeter groups, Invent. Math. 136 (1999) 323–351
  • P-E Caprace, B Mühlherr, Reflection rigidity of 2–spherical Coxeter groups, Proc. London Math. Soc. 94 (2007) 520–542
  • H S M Coxeter, The complete enumeration of finite groups of the form $R_i^2=(R_iR_j)^{k_{ij}}=1$, J. London Math. Soc. 10 (1935) 21–25
  • H S M Coxeter, Regular polytopes, third edition, Dover Publications, New York (1973)
  • V V Deodhar, On the root system of a Coxeter group, Comm. Algebra 10 (1982) 611–630
  • W N Franzsen, R B Howlett, Automorphisms of nearly finite Coxeter groups, Adv. Geom. 3 (2003) 301–338
  • W N Franzsen, R B Howlett, B Mühlherr, Reflections in abstract Coxeter groups, Comment. Math. Helv. 81 (2006) 665–697
  • M Mihalik, S Tschantz, Visual decompositions of Coxeter groups (2007)
  • B Mühlherr, The isomorphism problem for Coxeter groups, from: “The Coxeter legacy”, Amer. Math. Soc., Providence, RI (2006) 1–15
  • L Paris, Irreducible Coxeter groups, Internat. J. Algebra Comput. (to appear)
  • L Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976) 255–264
  • M Suzuki, Group theory I, Grundlehren der Mathematischen Wissenschaften 247, Springer, Berlin (1982)