# A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory: 401 (London Mathematical Society Lecture Note Series, Series Number 401)

Cambridge University Press, 12/6/2012

EAN 9781107608603, ISBN10: 1107608600

Paperback, 216 pages, 22.9 x 15.2 x 1.2 cm

Language: English

Originally published in English

The theory of SchurÃ¢â‚¬â€œWeyl duality has had a profound influence over many areas of algebra and combinatorics. This text is original in two respects: it discusses affine q-Schur algebras and presents an algebraic, as opposed to geometric, approach to affine quantum SchurÃ¢â‚¬â€œWeyl theory. To begin, various algebraic structures are discussed, including double RingelÃ¢â‚¬â€œHall algebras of cyclic quivers and their quantum loop algebra interpretation. The rest of the book investigates the affine quantum SchurÃ¢â‚¬â€œWeyl duality on three levels. This includes the affine quantum SchurÃ¢â‚¬â€œWeyl 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 RingelÃ¢â‚¬â€œHall algebra with a proof of the classical case. This text is ideal for researchers in algebra and graduate students who want to master RingelÃ¢â‚¬â€œHall algebras and SchurÃ¢â‚¬â€œWeyl duality.

Introduction

1. Preliminaries

2. Double RingelÃ¢â‚¬â€œHall algebras of cyclic quivers

3. Affine quantum Schur algebras and the SchurÃ¢â‚¬â€œWeyl reciprocity

4. Representations of affine quantum Schur algebras

5. The presentation and realization problems

6. The classical (v =1) case

Bibliography

Index.