On the Fully Commutative Elements of Coxeter Groups

Authors:
John R. Stembridge
Affiliations:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
1996

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Abstract

Let W be a Coxeter group. We define an element w ∈ 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.