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Authors
Bishop, M,Douglass, JM,Pfeiffer, G,Rohrle, G
Year
2013
Month
March
Journal
Journal Of Symbolic Computation
Title
Computations for Coxeter arrangements and Solomon's descent algebra: Groups of rank three and four
Status
Published
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()
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Coxeter group Descent algebra Orlik-Solomon algebra PARABOLIC SUBGROUPS SPECIAL INVOLUTIONS REFLECTION GROUPS CENTRALIZERS HYPERPLANES COHOMOLOGY COMPLEMENT
Volume
50
Issue
Start Page
139
End Page
158
Abstract
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group W afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of W. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of W related to the descent algebra of W. A consequence of our conjecture is that both the regular character of W and the character of the Orlik-Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of W, one for each conjugacy class of elements of W. The refined conjecture has been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjecture computationally for a given finite Coxeter group. We use these tools to verify the conjecture for all finite Coxeter groups of rank three and four, thus providing previously unknown decompositions of the regular characters and the Orlik-Solomon characters of these groups. (C) 2012 Elsevier B.V. All rights reserved.
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DOI 10.1016/j.jsc.2012.06.001
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