## On the fully commutative elements of Coxeter groups.(English)Zbl 0864.20025

Author’s abstract: Let $$W$$ be a Coxeter group. We define an element $$w\in W$$ to be fully commutative if any reduced expression for $$w$$ can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice and (4) the weak ordering and Bruhat ordering coincide.
Reviewer: L.Paris (Dijon)

### MSC:

 20F55 Reflection and Coxeter groups (group-theoretic aspects) 06A06 Partial orders, general 20F05 Generators, relations, and presentations of groups
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### References:

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