Three approaches to the Howe duality between quantum general linear supergroups

TL;DR

This paper presents two new approaches to establish Howe duality between quantum general linear supergroups, complementing the original method, and proves their equivalence through explicit action formulas.

Contribution

It introduces quantum differential operators and BLM realization methods as alternative approaches to Howe duality, demonstrating their equivalence.

Findings

Three approaches to Howe duality are shown to be equivalent.

Explicit action formulas establish the connection between methods.

New methods provide alternative frameworks for understanding quantum supergroup duality.

Abstract

The Howe duality between quantum general linear supergroups was firstly established by Y. Zhang via quantum coordinate superalgebras. In this paper, we provide two other approaches to this Howe duality. One is constructed by quantum differential operators, while the other is based on the Beilinson-Lusztig-MacPherson realization of $U_q(\mathfrak{gl}_{m|n})$. Moreover, we show that these three approaches are equivalent by giving their action formulas explicitly.