University of Minnesota Combinatorics Seminar: 2021-2022

Date: 09/17/2021

Speaker: Sarah Brauner

A Hyperoctahedral Analog of the Whitehouse Representation and Connections to the Type B Mantaci-Reutenauer Algebra
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 discovered by Whitehouse: that although they are defined as $S_n$ representations, they can also be understood via a "hidden" action of $S_{n+1}$. More surprisingly still, many of the connections between the Eulerian representations and objects such as configuration spaces and Solomon's descent algebra can be "lifted" to this family of $S_{n+1}$ representations, which we will call the Whitehouse representations. I will then discuss recent work generalizing the above scenario to the hyperoctahedral group, $B_n$. 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 (to be defined in the talk).

A Hyperoctahedral Analog of the Whitehouse Representation and Connections to the Type B Mantaci-Reutenauer Algebra

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 discovered by Whitehouse: that although they are defined as $S_n$ representations, they can also be understood via a "hidden" action of $S_{n+1}$. More surprisingly still, many of the connections between the Eulerian representations and objects such as configuration spaces and Solomon's descent algebra can be "lifted" to this family of $S_{n+1}$ representations, which we will call the Whitehouse representations. I will then discuss recent work generalizing the above scenario to the hyperoctahedral group, $B_n$. 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 (to be defined in the talk).