Physical Sciences and Mathematics Commons^™

On The Cyclically Fully Commutative Elements Of Coxeter Groups, T. Boothby, J. Burket, M. Eichwald, D. C. Ernst, R. M. Green, Matthew Macauley

Publications

Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is cyclically fully commutative (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink conversions. In this paper, we explore the combinatorics of the CFC elements and enumerate them in all Coxeter groups. …