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A Characterization of the Simply-Laced FC-Finite Coxeter Groups
Manabu Hagiwara1, Masao Ishikawaand2 and Hiroyuki Tagawa3
1Institute of Industrial Science, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan
manau@imailab.iis.u-tokyo.ac.jp
2Department of Mathematics, Tottori University, Koyama, Tottori 680-8551, Japan
ishikawa@fed.tottori-u.ac.jp
3Department of Mathematics, Wakayama University, Sakaedani, Wakayama 640-8510, Japan
tagawa@math.edu.wakayama-u.ac.jp
Annals of Combinatorics 8 (2) p.177-196 June, 2004
AMS Subject Classification: 06A07, 20F55
Abstract:
We call an element of a Coxeter group fully covering (or a fully covering element) if its length is equal to the number of the elements it covers in the Bruhat ordering. It is easy to see that the notion of fully covering is a generalization of the notion of a 321-avoiding permutation and that a fully covering element is a fully commutative element. Also, we call a Coxeter group bi-full if its fully commutative elements coincide with its fully covering elements. We show that the bi-full Coxeter groups are the ones of type An, Dn, En with no restriction on n. In other words, Coxeter groups of type E9, E10, ... are also bi-full. According to a result of Fan, a Coxeter group is a simply-laced FC-finite Coxeter group if and only if it is a bi-full Coxeter group.
Keywords: Coxeter groups, Bruhat ordering, fully commutative elements, fully covering elements

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