Quantum Schur Superalgebras and Kazhdan-Lusztig Combinatorics

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 $\text{sA}(m\mathrm{\mid }n)$. We also construct a cellular $\mathbb{Q}(\text{up})$-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 $\mathbb{Q}(\text{up})$ via super-cell modules.