Cohomology of Coxeter arrangements and Solomon's descent algebra

Details Cohomology of Coxeter arrangements and Solomon's descent algebra Cohomology of Coxeter arrangements and Solomon's descent algebra

2011-01-11

arxiv.org/abs/1101.2075v4

Mathematics

Representation Theory

29 pages; updated references

Scientific paper

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon 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 Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional 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 $A$.

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