Abstract
This paper provides a survey on the latest developments of $q$-Schur superalgebras, quantum linear supergroups and their canonical bases and irreducible (polynomial) representations. We first introduce the definitions of $q$-Schur superalgebras in three different contexts and their associated bases,and describe the relationships between the three bases. We then move on to display certain multiplication formulas in $q$-Schur superalgebras and discuss their applications to the new realizationof quantum linear supergroups and to the regular representation of a $q$-Schur superalgebra. We provide another application for the construction of canonical bases for the positive part of thequantum linear supergroup. As a by-product, a semi-simplicity criterion is given for $q$-Schur superalgebras. Moreover, by generalising the idea of Alperins weight conjecture and Scottspermutation representation theory, we provide a classification of irreducible modules for a $q$-Schur superalgebra. We also mention a new approach to introduce infinitesimal and little $q$-Schur superalgebras without using quantum coordinate superalgebras.
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