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When is a Coxeter System Determined by its Coxeter Group?
Author(s) -
Charney Ruth,
Davis Michael
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008583
Subject(s) - coxeter group , longest element of a coxeter group , mathematics , combinatorics , coxeter element , point group , coxeter complex , group (periodic table) , section (typography) , set (abstract data type) , artin group , physics , computer science , quantum mechanics , programming language , operating system
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 W =〈 S |( st ) m ( s , t ) 〉 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.