A Double Hall Algebra Approach to Affine Quantum Schur--Weyl Theory

Abstract

We investigate the structure of the double Ringel-Hall algebras associated with cyclic quivers and its connections with quantum loop algebras of ${\mathfrak{gl}}_n$, affine quantum Schur algebras and affine Hecke algebras. This includes their Drinfeld-Jimbo type presentation, affine quantum Schur-Weyl reciprocity, representations of affine quantum Schur algebras, and connections with various existing works by Lusztig, Varagnolo-Vasserot, Schiffmann, Hubery, Chari-Pressley, Frenkel-Mukhin, etc. We will also discuss conjectures on a realization of Beilinson-Lusztig-MacPherson type and Lusztig type integral forms for double Ringel-Hall algebras.

166 pages

References in corpus (7)

• The q-characters of representations of quantum affine algebras and deformations of W-algebras
• Quantum Affine Algebras and their Representations
• From double affine Hecke algebras to quantized affine Schur algebras
• Presenting Schur Algebras
• On the affine Schur algebra of type A
• On the Comultiplication in Quantum Affine Algebras
• Quantum affine algebras and affine Hecke algebras

Cited by in corpus (8)

• Quiver Schur algebras and q-Fock space
• Semi-derived Ringel-Hall algebras and Drinfeld double
• BLM realization for the integral form of quantum ${\mathfrak{gl}}_n$
• Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality
• Quantum Schur duality of affine type C with three parameters
• Affine quantum Schur algebras at roots of unity
• Integral affine Schur-Weyl reciprocity
• Multiplication formulas and semisimplicity for q-Schur superalgebras