A Characterization of the Simply-Laced FC-Finite
Coxeter Groups |
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Authors: | Manabu Hagiwara Masao Ishikawa and Hiroyuki Tagawa |
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Institution: | (1) Institute of Industrial Science, University of Tokyo, Meguro-ku, 153-8505 Tokyo, Japan;(2) Department of Mathematics, Tottori University, Koyama, 680-8551 Tottori, Japan;(3) Department of Mathematics, Wakayama University, Sakaedani, 640-8510 Wakayama, Japan |
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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.AMS Subject Classification: 06A07, 20F55. |
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Keywords: | Coxeter groups Bruhat ordering fully commutative elements fully covering elements |
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