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

Book Description

The theory of Schur–Weyl 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 Schur–Weyl theory. To begin, various algebraic structures are discussed, including double Ringel–Hall algebras of cyclic quivers and their quantum loop algebra interpretation. The rest of 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 ideal for researchers in algebra and graduate students who want to master Ringel–Hall algebras and Schur–Weyl duality.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Dedication Page
  5. Contents
  6. Introduction
  7. 1 Preliminaries
    1. 1.1 The loop algebra gÎn and some notation
    2. 1.2 Representations of cyclic quivers and Ringel–Hall algebras
    3. 1.3 The quantum loop algebra U(sÎn)
    4. 1.4 Three types of generators and associated monomial bases
    5. 1.5 Hopf structure on extended Ringel–Hall algebras
  8. 2 Double Ringel–Hall algebras of cyclic quivers
    1. 2.1 Drinfeld doubles and the Hopf algebra D∆(n)
    2. 2.2 Schiffmann–Hubery generators
    3. 2.3 Presentation of D∆(n)
    4. 2.4 Some integral forms
    5. 2.5 The quantum loop algebra U(gÎn)
    6. 2.6 Semisimple generators and commutator formulas
  9. 3 Affine quantum Schur algebras and the Schur–Weyl reciprocity
    1. 3.1 Cyclic flags: the geometric definition
    2. 3.2 Affine Hecke algebras of type A: the algebraic definition
    3. 3.3 The tensor space interpretation
    4. 3.4 BLM bases and multiplication formulas
    5. 3.5 The D∆(n)-H∆(r)-bimodule structure on tensor spaces
    6. 3.6 A comparison with the Varagnolo–Vasserot action
    7. 3.7 Triangular decompositions of affine quantum Schur algebras
    8. 3.8 Affine quantum Schur–Weyl duality, I
    9. 3.9 Polynomial identity arising from semisimple generators
    10. 3.10 Appendix
  10. 4 Representations of affine quantum Schur algebras
    1. 4.1 Affine quantum Schur–Weyl duality, II
    2. 4.2 Chari–Pressley category equivalence and classification
    3. 4.3 Classification of simple S∆(n, r)ℂ-modules: the upward approach
    4. 4.4 Identification of simple S∆(n, r)ℂ-modules: the n > r case
    5. 4.5 Application: the set lr(n)
    6. 4.6 Classification of simple S∆(n, r)ℂ-modules: the downward approach
    7. 4.7 Classification of simple U∆(n, r)ℂ-modules
  11. 5 The presentation and realization problems
    1. 5.1 McGerty’s presentation for U∆(n, r)
    2. 5.2 Structure of affine quantum Schur algebras
    3. 5.3 Presentation of S∆(r, r)
    4. 5.4 The realization conjecture
    5. 5.5 Lusztig’s transfer maps on semisimple generators
  12. 6 The classical (v = 1) case
    1. 6.1 The universal enveloping algebra U(gÎn)
    2. 6.2 More multiplication formulas in affine Schur algebras
    3. 6.3 Proof of Conjecture 5.4.2 at v = 1
    4. 6.4 Appendix: Proof of Proposition 6.2.3
  13. Bibliography
  14. Index