Schur-Weyl duality for orthogonal groups

TL;DR

This paper establishes a duality between the Brauer algebra and orthogonal groups over infinite fields of odd characteristic, providing new insights into their algebraic structures and representations.

Contribution

It proves Schur-Weyl duality for orthogonal groups and Brauer algebras over arbitrary infinite fields of odd characteristic, and describes the annihilator of tensor space explicitly.

Findings

Schur-Weyl duality holds between Brauer algebra and orthogonal group.

Connected components of the orthogonal monoid are normal varieties when m is even.

Explicit, characteristic-free description of the annihilator of tensor space in the Brauer algebra.

Abstract

We prove Schur--Weyl duality between the Brauer algebra $\mathfrak{B}_n(m)$ and the orthogonal group $O_{m}(K)$ over an arbitrary infinite field $K$ of odd characteristic. If $m$ is even, we show that each connected component of the orthogonal monoid is a normal variety; this implies that the orthogonal Schur algebra associated to the identity component is a generalized Schur algebra. As an application of the main result, an explicit and characteristic-free description of the annihilator of $n$-tensor space $V^{\otimes n}$ in the Brauer algebra $mathfrak{B}_n(m)$ is also given.