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Bounded Littlewood identities
About this Title
Eric M. Rains and S. Ole Warnaar
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 270, Number 1317
ISBNs: 978-1-4704-4690-1 (print); 978-1-4704-6522-3 (online)
DOI: https://doi.org/10.1090/memo/1317
Published electronically: May 14, 2021
Keywords: Macdonald–Koornwinder polynomials,
Hall–Littlewood polynomials,
virtual Koornwinder integrals,
character formulas,
Rogers–Ramanujan identities,
plane partitions
Table of Contents
Chapters
- Acknowledgements
- 1. Introduction
- 2. Macdonald–Koornwinder theory
- 3. Virtual Koornwinder integrals
- 4. Bounded Littlewood identities
- 5. Applications
- 6. Open problems
- A. The Weyl–Kac formula
- B. Limits of elliptic hypergeometric integrals
Abstract
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. These identities, which take the form of decomposition formulas for Macdonald polynomials of type $(R,S)$ in terms of ordinary Macdonald polynomials, are $q,t$-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon’s famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of $\big (\mathrm {GL}(n,\mathbb {R}),\mathrm {O}(n)\big )$ as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers–Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko–Macdonald-type basic hypergeometric series.- George E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 4082–4085. MR 351985, DOI 10.1073/pnas.71.10.4082
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