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WHEN IS A COXETER SYSTEM DETERMINED BY ITS COXETER GROUP?

Published online by Cambridge University Press:  01 April 2000

RUTH CHARNEY  and
MICHAEL DAVIS
RUTH CHARNEY
Affiliation:
Department of Mathematics, Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210, USA; charney@math.ohio-state.edu, mdavis@math.ohio-state.edu
MICHAEL DAVIS
Affiliation:
Department of Mathematics, Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210, USA; charney@math.ohio-state.edu, mdavis@math.ohio-state.edu
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Abstract

A Coxeter system is a pair (W, S) where W is a group and where S is a set of involutions in W such that W has a presentation of the form

formula here

Here m(s, t) denotes the order of st in W and in the presentation for W, (s, t) ranges over all pairs in S × S such that m(s, t) ≠ ∞. We further require the set S to be finite. W is a Coxeter group and S is a fundamental set of generators for W.

Obviously, if S is a fundamental set of generators, then so is wSw−1, for any w∈W. Our main result is that, under certain circumstances, this is the only way in which two fundamental sets of generators can differ. In Section 3, we will prove the following result as Theorem 3.1.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 441 - 461
Copyright
The London Mathematical Society 2000

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