• DOI:10.1112/jlms.12519
• Corpus ID: 218596316

Eulerian representations for real reflection groups

@article{Brauner2022EulerianRF,
  title={Eulerian representations for real reflection groups},
  author={Sarah Brauner},
  journal={Journal of the London Mathematical Society},
  year={2022},
  volume={105}
}

• 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$B_{n}$ , H3$H_{3}$ or the dihedral group I2(m)$I_{2}(m)$ — by giving six characterizations of the Eulerian representations, including as components of the associated graded Varchenko–Gelfand ring V$\mathcal {V}$ . As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with a fixed number of descents if and only if W$W$ is coincidental. More generally, for any finite real reflection group, we give a case‐free construction of a family of Eulerian representations described by a flat decomposition of the ring V$\mathcal {V}$ .