A note on Schur-Weyl dualities for $GL(m)$ and $GL(m|n)$

TL;DR

This paper provides a unified elementary proof of the second part of classical and super Schur-Weyl dualities for general linear groups and supergroups over infinite fields, extending results to positive characteristic.

Contribution

It introduces the second part of mixed and mixed super Schur-Weyl dualities over infinite fields of positive characteristic, broadening the scope of existing duality theories.

Findings

Proves the second part of mixed Schur-Weyl dualities in positive characteristic.

Establishes the endomorphism algebras for tensor and mixed tensor spaces.

Extends classical dualities to supergroups and positive characteristic fields.

Abstract

We use a unified elementary approach to prove the second part of classical, mixed, super, and mixed super Schur-Weyl dualities for general linear groups and supergroups over an infinite ground field of arbitrary characteristic. These dualities describe the endomorphism algebras of the tensor space and mixed tensor space, respectively, over the group algebra of the symmetric group and the Brauer wall algebra, respectively. Our main new results are the second part of the mixed Schur-Weyl dualities and mixed super Schur-Weyl dualities over an infinite ground field of positive characteristic.