# Cohomology of Coxeter arrangements and Solomon's descent algebra

@article{Douglass2011CohomologyOC, title={Cohomology of Coxeter arrangements and Solomon's descent algebra}, author={J. Matthew Douglass and Goetz Pfeiffer and Gerhard R{\"o}hrle}, journal={arXiv: Representation Theory}, year={2011} }

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…

## 11 Citations

COMPUTATIONS FOR COXETER ARRANGEMENTS AND SOLOMON’S DESCENT ALGEBRA II: GROUPS OF RANK FIVE AND SIX

- Mathematics
- 2018

. In recent papers we have reﬁned a conjecture of Lehrer and Solomon expressing the character of a ﬁnite Coxeter group W acting on the graded components of its Orlik-Solomon algebra as a sum of…

Title Computations for Coxeter arrangements and Solomon ' s descentalgebra II : Groups of rank five and six

- Mathematics
- 2019

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the graded components of its Orlik-Solomon algebra as a sum of…

Computations for Coxeter arrangements and Solomon's descent algebra: Groups of rank three and four

- MathematicsJ. Symb. Comput.
- 2013

A decomposition of the group algebra of a hyperoctahedral group

- Mathematics
- 2016

The descent algebra of a finite Coxeter group W is a subalgebra of the group algebra defined by Solomon. Descent algebras of symmetric groups have properties that are not shared by other Coxeter…

Title Computations for Coxeter arrangements and Solomon ' s descentalgebra III : Groups of rank seven and eight

- Mathematics
- 2018

In this paper we extend the computations in parts I and II of this series of papers and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W…

A decomposition of the group algebraof a hyperoctahedral group

- Mathematics
- 2018

The descent algebra of a finite Coxeter group W is a subalgebra of the group algebra defined by Solomon. Descent algebras of symmetric groups have properties that are not shared by other Coxeter…

Configuration spaces of points in an elliptic curve

- Mathematics
- 2018

In this work we study elliptic arrangements and, in particular, the case of ordered configuration spaces of points. We focus on the cohomology algebra of the configuration and its dependency on the…

The connection between combinatorics and cohomology of elliptic arrangements

- Mathematics
- 2018

We prove that the cohomology algebra of elliptic arrangements depends only on the poset of layers. In the particular case of braid elliptic arrangements, we study the cohomology as representation and…

Computations for Coxeter arrangements and Solomon's descent algebra III: Groups of rank seven and eight

- Mathematics
- 2013

Schur-Weyl duality and the free Lie algebra

- Mathematics
- 2015

We prove an analogue of Schur-Weyl duality for the space of homogeneous Lie polynomials of degree r in n variables.

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Computations for Coxeter arrangements and Solomon's descent algebra: Groups of rank three and four

- MathematicsJ. Symb. Comput.
- 2013

An inductive approach to Coxeter arrangements and Solomon’s descent algebra

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In our recent paper (Douglass et al. arXiv:1101.2075 (2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik–Solomon algebra of W can be decomposed into a sum…

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Let W be a finite Coxeter group, realized as a group generated by reflections in the /-dimensional Euclidean space V. Let s/ be the hyperplane arrangement in C* = F(g)RC consisting of the…