A Type B analog of the Whitehouse representation

TL;DR

This paper introduces a new family of $B_n$-representations that extend Whitehouse's Eulerian representation lifts from $S_n$ to $S_{n+1}$, with combinatorial, topological, and algebraic interpretations.

Contribution

It provides a Type B analog of Whitehouse's lifts, including combinatorial, topological, and algebraic frameworks, extending the understanding of Eulerian representations.

Findings

Constructed $B_n$-representations lifting to $B_{n+1}$

Connected representations to cohomology of $bZ_2$-orbit configuration spaces

Established properties analogous to Type A case, including equivariant cohomology and Varchenko-Gelfand ring

Abstract

We give a Type $B$ analog of Whitehouse's lifts of the Eulerian representations from $S_n$ to $S_{n+1}$ by introducing a family of $B_{n}$-representations that lift to $B_{n+1}$. As in Type $A$, we interpret these representations combinatorially via a family of orthogonal idempotents in the Mantaci-Reutenauer algebra, and topologically as the graded pieces of the cohomology of a certain $\mathbb{Z}_{2}$-orbit configuration space of $\mathbb{R}^{3}$. We show that the lifted $B_{n+1}$-representations also have a configuration space interpretation, and further parallel the Type $A$ story by giving analogs of many of its notable properties, such as connections to equivariant cohomology and the Varchenko-Gelfand ring.