Bounded Littlewood identities

We describe a method, based on the theory of Macdonald–Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald’s partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. Thes...

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Bibliographic Details
Main Authors: Rains, Eric M., Warnaar, S. Ole
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2021
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Combinatorial identities
Orthogonal polynomials
plane partitions
virtual Koornwinder integrals
Rogers–Ramanujan identities
Macdonald–Koornwinder polynomials
Hall–Littlewood polynomials
character formulas
ISBN:1470446901, 9781470446901
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Table of Contents:
  • Acknowledgements -- Introduction -- Macdonald–Koornwinder theory -- Virtual Koornwinder integrals -- Bounded Littlewood identities -- Applications -- Open problems -- The Weyl–Kac formula -- Limits of elliptic hypergeometric integrals
  • Cover -- Title page -- Acknowledgements -- Chapter 1. Introduction -- 1.1. Littlewood identities -- 1.2. Outline -- Chapter 2. Macdonald-Koornwinder theory -- 2.1. Partitions -- 2.2. Generalised -shifted factorials -- 2.3. Rogers-Szegő polynomials -- 2.4. Plethystic notation -- 2.5. Macdonald polynomials -- 2.6. Koornwinder polynomials -- 2.7. Macdonald-Koornwinder polynomials -- 2.8. Hall-Littlewood polynomials -- Chapter 3. Virtual Koornwinder integrals -- 3.1. Basic definitions -- 3.2. Closed-form evaluations-the Macdonald case -- 3.3. Closed-form evaluations-the Hall-Littlewood case -- Chapter 4. Bounded Littlewood identities -- 4.1. Statement of results -- 4.2. Proofs of Theorems 4.1-4.8 -- Chapter 5. Applications -- 5.1. Plane partitions -- 5.2. Character identities for affine Lie algebras -- 5.3. Rogers-Ramanujan identities -- 5.4. Quadratic transformations for Kaneko-Macdonald-type basic hypergeometric series -- Chapter 6. Open problems -- 6.1. Missing -analogues -- 6.2. Littlewood identities for near-rectangular partitions -- 6.3. Littlewood identities of Pfaffian type -- 6.4. Elliptic Littlewood identities -- 6.5. , -Littlewood-Richardson coefficients -- 6.6. Dyson-Macdonald-type identities -- Appendix A. The Weyl-Kac formula -- Appendix B. Limits of elliptic hypergeometric integrals -- Bibliography -- Back Cover