Cohomology of Coxeter arrangements and Solomon's descent algebra
Abstract
We refine a conjecture by Lehrer and Solomon on the structure of the OrlikSolomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the OrlikSolomon algebra of $W$ can be decomposed into a sum of induced onedimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_L$ is a parabolic subgroup of $W$ all of whose irreducible factors are of type $A$.
 Publication:

arXiv eprints
 Pub Date:
 January 2011
 arXiv:
 arXiv:1101.2075
 Bibcode:
 2011arXiv1101.2075D
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 20F55;
 05E10;
 52C35
 EPrint:
 31 pages