Quantum Schur Superalgebras and Kazhdan-Lusztig Combinatorics

TL;DR

This paper introduces quantum Schur superalgebras, exploring their properties, bases, and cell structures, and classifies their irreducible representations using super-cell modules, extending classical combinatorics to the super setting.

Contribution

It defines quantum Schur superalgebras, constructs their cellular bases, and classifies irreducible representations via super-cell modules, linking to super-Knuth correspondences.

Findings

Established base change and duality properties.

Constructed cellular bases and identified super-cells.

Classified all irreducible representations over Q(6)

Abstract

We introduce the notion of quantum Schur (or $q$-Schur) superalgebras. These algebras share certain nice properties with $q$-Schur algebras such as base change property, existence of canonical $\mathbb Z[v,v^{-1}]$-bases, and the duality relation with quantum matrix superalgebra $\sA(m|n)$. We also construct a cellular $\mathbb Q(\up)$-basis and determine its associated cells, called super-cells, in terms of a Robinson--Schensted--Knuth super-correspondence. In this way, we classify all irreducible representations over $\mathbb Q(\up)$ via super-cell modules.