A double Hall algebra approach to affine quantum Schur-Weyl theory.
The theory of Schur-Weyl duality has had a profound influence over many areas of algebra and combinatorics. This text discusses affine \(q\)-Schur algebras and presents an algebraic, as opposed to geometric, approach to affine quantum Schur-Weyl theory. Various algebraic structures are discussed, including double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. The book investigates the affine quantum Schur-Weyl duality on three levels. This includes the affine quantum Schur-Weyl reciprocity, the bridging role of affine \(q\)-Schur algebras between representations of the quantum loop algebras and those of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel-Hall algebra with a proof of the classical case. -- This text is very useful for researchers in algebra and graduate students who want to master Ringel-Hall algebras and Schur-Weyl duality.
scientific article; zbMATH DE number 6059853