A Type B analog of the Whitehouse representation
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 MantaciReutenauer 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 VarchenkoGelfand ring.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.09504
 arXiv:
 arXiv:2203.09504
 Bibcode:
 2022arXiv220309504B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Topology;
 Mathematics  Representation Theory;
 05Exx;
 20F55;
 55R80;
 55N91;
 14N20;
 52C35
 EPrint:
 final version, to appear in Mathematische Zeitschrift