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Proof of some Littlewood identities conjectured by Lee, Rains and Warnaar

Seamus P. Albion

Abstract

We prove a novel pair of Littlewood identities for Schur functions, recently conjectured by Lee, Rains and Warnaar in the Macdonald case, in which the sum is over partitions with empty 2-core. As a byproduct we obtain a new Littlewood identity in the spirit of Littlewood's original formulae.

Proof of some Littlewood identities conjectured by Lee, Rains and Warnaar

Abstract

We prove a novel pair of Littlewood identities for Schur functions, recently conjectured by Lee, Rains and Warnaar in the Macdonald case, in which the sum is over partitions with empty 2-core. As a byproduct we obtain a new Littlewood identity in the spirit of Littlewood's original formulae.

Paper Structure

This paper contains 11 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Partitions and ($\mathrm{BC}_n$)-symmetric functions
  3. Partitions
  4. Auxiliary results
  5. Schur functions
  6. Koornwinder polynomials and integrals
  7. Vanishing integrals
  8. Bounded Littlewood identities
  9. A bounded analogue of Theorem \ref{['Thm_q=t']}
  10. Proof of Theorem \ref{['Thm_q=t']}
  11. Proof of Corollary \ref{['Cor']}

Key Result

Theorem 1.1

As identities in $\hat{\Lambda}_{\mathbb{Q}(q)}$ at the alphabet $x=(x_1,x_2,x_3,\dots)$ we have that and

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4: RW21
  • Proposition 3.1: LRW20
  • Proposition 3.2
  • ...and 3 more