Enhanced Brauer algebras and enhanced dualities for orthogonal and symplectic groups

TL;DR

This paper extends the concept of enhanced dualities from type A to classical groups of types B, C, and D by introducing an enhanced Brauer algebra and exploring related dualities.

Contribution

It introduces an enhanced Brauer algebra and establishes new dualities for orthogonal and symplectic groups, expanding the framework of enhanced dualities beyond type A.

Findings

Established restricted and Levi dualities for classical groups

Described parabolic duality through explicit calculations

Connected enhanced Brauer algebra with classical group dualities

Abstract

This note is a sequel to Shu-Xue-Yao's paper \cite{BYY} where the author studied the so-called enhanced groups and related dualities for type $A$. In this note, we continue to investigate the enhanced dualities for classical groups of type $B$, $C$, and $D$. To show this, we first introduce an enhanced Brauer algebra. Due to Proposition \ref{B dual}, we can easily obtain the restricted and Levi dualities, hence the parabolic duality is naturally described. Moreover, a special case is listed in \S\ref{3} by some explicit calculations.