Eulerian representations for real reflection groups

Abstract

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 $A$, the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. In this paper, we unify these results and show that they hold for any reflection group of coincidental type -- that is, $S_{n}$, $B_{n}$, $H_{3}$ or the dihedral group $I_{2}(m)$ -- by giving six characterizations of the Eulerian representations, including as components of the associated graded of the Varchenko-Gelfand ring $\mathcal{V}$. As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with the same number of descents if and only if $W$ is coincidental. More generally, when $W$ is 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 $\mathcal{V}$.