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Algebra of operators for Q-Schur polynomials

Nikita Tselousov

TL;DR

This work constructs operator algebras acting on Schur and Q-Schur polynomials, linking KP/BKP integrability to Yangian-like symmetries. The Schur sector is realized as a degenerate affine Yangian of gl1 with ε1=1, ε2=−1, yielding commuting subalgebras generated by ψ_a and by integer-ray families, and enabling reconstruction of Schur polynomials from the first commutative family via box-addition rules. The Q-Schur sector extends the construction to BKP integrability using ladder diagrams and a U-operator, producing analogous diagonal operators Ψ_a with box-additivity and single-hook expansions, and introducing explicit Ω_a(x) governing their action. Collectively, the paper provides explicit operator realizations, recursive structures, and closed-form diagonal actions for both Schur and Q-Schur bases, connecting to Calogero-type integrable systems and Macdonald theory through a unified framework of affine Yangian–like algebras.

Abstract

We consider algebras acting on Schur and Q-Schur polynomials, corresponding to Kadomtsev-Petviashvili (KP) and BKP hierarchies. We present them in the spirit of affine Yangians, paying special attention to commutative subalgebras, box additivity property of eigenvalues and single hook expansion of operators.

Algebra of operators for Q-Schur polynomials

TL;DR

This work constructs operator algebras acting on Schur and Q-Schur polynomials, linking KP/BKP integrability to Yangian-like symmetries. The Schur sector is realized as a degenerate affine Yangian of gl1 with ε1=1, ε2=−1, yielding commuting subalgebras generated by ψ_a and by integer-ray families, and enabling reconstruction of Schur polynomials from the first commutative family via box-addition rules. The Q-Schur sector extends the construction to BKP integrability using ladder diagrams and a U-operator, producing analogous diagonal operators Ψ_a with box-additivity and single-hook expansions, and introducing explicit Ω_a(x) governing their action. Collectively, the paper provides explicit operator realizations, recursive structures, and closed-form diagonal actions for both Schur and Q-Schur bases, connecting to Calogero-type integrable systems and Macdonald theory through a unified framework of affine Yangian–like algebras.

Abstract

We consider algebras acting on Schur and Q-Schur polynomials, corresponding to Kadomtsev-Petviashvili (KP) and BKP hierarchies. We present them in the spirit of affine Yangians, paying special attention to commutative subalgebras, box additivity property of eigenvalues and single hook expansion of operators.

Paper Structure

This paper contains 13 sections, 94 equations, 2 figures.

Table of Contents

  1. Introduction and discussion
  2. Schur case
  3. Basic formulas
  4. $W_{1+\infty}$ algebra as degenerate affine Yangian $\mathfrak{gl}_1$
  5. Commutative integer rays of operators
  6. Schur polynomials from the first commutative family
  7. Explicit form of operators $\hat{\psi}_a$
  8. Q-Schur case
  9. Basic formulas
  10. Towards the algebra
  11. Commutative integer rays of operators
  12. Q-Schur polynomials from the first commutative family
  13. Explicit form of operators $\hat{\Psi}_a$

Figures (2)

  • Figure 1: Example of Young diagram $\lambda = [13,10,9,7,6,5,3,3,2,2,1]$. In our notation $\lambda_i$ is the length of $i$-th row counting from the bottom. Young diagram $\lambda$ can be considered as a way of tight packing of $|\lambda| = \sum_{i} \lambda_i$ identical boxes in the corner. Gray dots correspond to the set of possible positions for the new boxes -- i.e. $\text{Add}(\lambda)$. By the sign "$\times$" we mark boxes that can be removed from the diagram $\lambda$, i.e. the set $\text{Rem}(\lambda)$.
  • Figure 2: Example of ladder Young diagram $\lambda = [15,12,10,7,4,3,2]$. In our notation $\lambda_i$ is the length of $i$-th row counting from the bottom. Ladder Young diagram $\lambda$ can be considered as a way of tight packing of $|\lambda| = \sum_{i} \lambda_i$ identical boxes under an infinite ladder. Gray dots correspond to the set of possible positions for the new boxes, i.e. $\text{Add}(\lambda)$. By the sign "$\times$" we mark boxes that can be removed from the diagram $\lambda$, i.e. the set $\text{Rem}(\lambda)$.