Séminaire Lotharingien de Combinatoire, 85B.15 (2021), 12 pp.
Eulerian Representations for Real Reflection Groups
The Eulerian idempotents of a real reflection group W generate a family of W-representations decomposing the regular representation, called the Eulerian representations. In Type A, the Eulerian representations are well-studied and 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, Bn, H3 or the dihedral group I2(m) - by giving six characterizations of the Eulerian representations, including as components of the associated graded Varchenko--Gelfand ring V. As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with the same descent number if and only if W is coincidental.
More generally, when W is any real, finite reflection group, we give a case-free construction of a family of Eulerian representations described by a flat-decomposition of the ring V.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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