Eulerian representations for real reflection groups

  title={Eulerian representations for real reflection groups},
  author={Sarah Brauner},
  journal={Journal of the London Mathematical Society},
  • Sarah Brauner
  • Published 12 May 2020
  • Mathematics
  • Journal of the London Mathematical Society
The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type A$A$ , the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. Here, we unify these results and show that they hold for any reflection group of coincidental type — that is, Sn$S_{n}$ , Bn… 
A Type B analog of the Whitehouse representation
We give a Type B analog of Whitehouse’s lifts of the Eulerian representations from Sn to Sn+1 by introducing a family of Bn-representations that lift to Bn+1. As in Type A, we interpret these


Invariant theory for coincidental complex reflection groups
V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and he
Cohomology of Coxeter arrangements and Solomon's descent algebra
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
Non-broken circuits of reflection groups and factorization inDn
AbstractThe set of non-broken circuits of a reflection group W, denoted NBC(W), appears as a basis of the Orlik-Solomon algebra ofW. The factorization of their enumerating polynomial $$\sum S \in
Spectra of Symmetrized Shuffling Operators
(Abridged abstract) For a finite real reflection group W and a W-orbit O of flats in its reflection arrangement---or equivalently a conjugacy class of its parabolic subgroups---we introduce a
Foulkes characters for complex reflection groups
Foulkes discovered a marvelous set of characters for the symmetric group by summing Specht modules of certain ribbon shapes according to height. These characters have many remarkable properties and
The $(1-\mathbb{E})$-transform in combinatorial Hopf algebras
We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a “transformation of alphabets”,
Hyperoctahedral Operations on Hochschild Homology
Abstract We construct Families {ρ l, k n } of orthogonal idempotents of the hyperoctahedral group algebras Q [ B n ], which commute with the Hochschild boundary operators b n =∑ n i=0 (−1) i d i . We
A decomposition of Solomon's descent algebra
On a curious variant of the Sn-module Lien
We introduce a variant of the much-studied Lie representation of the symmetric group Sn, which we denote by Lie n . Our variant gives rise to a decomposition of the regular representation as a sum of
A hyperoctahedral analogue of the free lie algebra