Canonical bases for the quantum linear supergroups

TL;DR

This paper constructs combinatorial canonical bases for quantum superalgebras of type gl_{m|n} and explores their connections with Kazhdan-Lusztig bases in quantum Schur superalgebras, extending to polynomial representations.

Contribution

It provides a new combinatorial construction of canonical bases for quantum superalgebras and relates them to existing Kazhdan-Lusztig bases, extending the framework to polynomial representations.

Findings

Constructed combinatorial canonical bases for quantum superalgebras.

Established relationships between these bases and Kazhdan-Lusztig bases.

Extended the relationship to simple polynomial representations.

Abstract

We give a combinatorial construction for the canonical bases of the $\pm$-parts of the quantum enveloping superalgebra $\bfU(\mathfrak{gl}_{m|n})$ and discuss their relationship with the Kazhdan-Lusztig bases for the quantum Schur superalgebras $\bsS(m|n,r)$ introduced in \cite{DR}. We will also extend this relationship to the induced bases for simple polynomial representations of $\bfU(\mathfrak{gl}_{m|n})$.