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HOMOTOPY TYPE OF THE BOOLEAN COMPLEX OF A COXETER SYSTEM

By Kári Ragnarsson and Bridget Eileen Tenner

Abstract

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, this boolean complex is homotopy equivalent to a wedge of (n − 1)-dimensional spheres, and the number of such spheres can be computed recursively. Specific calculations of this number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group

Year: 2008
OAI identifier: oai:CiteSeerX.psu:10.1.1.311.5988
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