Discrete Geometry and Combinatorics Seminar
Abstract: The Eulerian idempotents of the symmetric group and the representations they generate, called the Eulerian representations, are a topic of long-standing interest to representation theorists, combinatorialists and topologists. In this talk, I will focus on a property of the Eulerian representations first studied by Whitehouse: that although the Eulerian representations are defined as Sn representations, they can also be understood via a "hidden" action of Sn+1. More surprisingly still, many of the connections between the Eulerian representations and configuration spaces, equivariant cohomology, and Solomon's descent algebra can be "lifted" to this family of Sn+1 representations, which we will call the Whitehouse representations. I will then discuss recent work generalizing the above scenario to the hyperoctahedral group, Bn. In this setting, configuration spaces will be replaced by certain orbit configuration spaces and Solomon's descent algebra is replaced by the Type B Mantaci-Reutenauer algebra. All of the above will be defined in the talk.