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Quantum Littlewood correspondences

Naihuan Jing, Yinlong Liu, Jian Zhang

TL;DR

The paper develops a quantum generalization of Littlewood correspondences for immanants by defining quantum immanants in the quantum coordinate algebra $A_q(\mathrm{Mat}_n)$ via Hecke algebra representations. It establishes a quantum Kostant-type trace identity through Schur–Weyl–Jimbo duality, and derives quantum analogs of Littlewood correspondences I–III, including $q$-Goulden–Jackson identities and $q$-Littlewood–Merris–Watkins relations. A central construct is the Bethe subalgebra $\mathfrak{R}_n$, generated by commutative elementary symmetric-like elements $\alpha_k$, which is isomorphic to coinvariants of symmetric functions and connects to a symmetric-function map $\Phi$. The results unify representation-theoretic and combinatorial aspects in the quantum setting, providing a robust framework for $q$-immanants, dualities, and associated identities with potential applications in quantum groups and noncommutative symmetric function theory.

Abstract

In the 1940s Littlewood formulated three fundamental correspondences for the immanants and Schur symmetric functions on the general linear group, which establish deep connections between representation theory of the symmetric group and the general linear group parallel to the Schur-Weyl duality. In this paper, we introduce the notion of quantum immanants in the quantum coordinate algebra using primitive idempotents of the Hecke algebra. By employing $R$-matrix techniques, we establish the quantum analog of Littlewood correspondences between quantum immanants and Schur functions for the quantum coordinate algebra. In the setting of the Schur-Weyl-Jimbo duality, we construct an exact correspondence between the Gelfand-Tsetlin bases of the irreducible representations of the quantum enveloping algebra $U_q(\mathfrak{gl}(n))$ and Young's orthonormal basis of an irreducible representation of the Hecke algebra $\mathcal H_m$. This isomorphism leads to our trace formula for the quantum immanants, which settled the generalization problem of $q$-analog of Kostant's formular for $λ$-immanants. As applications, we also derive general $q$-Littlewood-Merris-Watkins identities and $q$-Goulden-Jackson identities as special cases of the quantum Littlewood correspondence III.

Quantum Littlewood correspondences

TL;DR

The paper develops a quantum generalization of Littlewood correspondences for immanants by defining quantum immanants in the quantum coordinate algebra via Hecke algebra representations. It establishes a quantum Kostant-type trace identity through Schur–Weyl–Jimbo duality, and derives quantum analogs of Littlewood correspondences I–III, including -Goulden–Jackson identities and -Littlewood–Merris–Watkins relations. A central construct is the Bethe subalgebra , generated by commutative elementary symmetric-like elements , which is isomorphic to coinvariants of symmetric functions and connects to a symmetric-function map . The results unify representation-theoretic and combinatorial aspects in the quantum setting, providing a robust framework for -immanants, dualities, and associated identities with potential applications in quantum groups and noncommutative symmetric function theory.

Abstract

In the 1940s Littlewood formulated three fundamental correspondences for the immanants and Schur symmetric functions on the general linear group, which establish deep connections between representation theory of the symmetric group and the general linear group parallel to the Schur-Weyl duality. In this paper, we introduce the notion of quantum immanants in the quantum coordinate algebra using primitive idempotents of the Hecke algebra. By employing -matrix techniques, we establish the quantum analog of Littlewood correspondences between quantum immanants and Schur functions for the quantum coordinate algebra. In the setting of the Schur-Weyl-Jimbo duality, we construct an exact correspondence between the Gelfand-Tsetlin bases of the irreducible representations of the quantum enveloping algebra and Young's orthonormal basis of an irreducible representation of the Hecke algebra . This isomorphism leads to our trace formula for the quantum immanants, which settled the generalization problem of -analog of Kostant's formular for -immanants. As applications, we also derive general -Littlewood-Merris-Watkins identities and -Goulden-Jackson identities as special cases of the quantum Littlewood correspondence III.
Paper Structure (14 sections, 21 theorems, 171 equations)

This paper contains 14 sections, 21 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. The quantum coordinate algebra and the Hecke algebra
  3. Quantum algebra $A_{q}(\mathrm{Mat}_{n})$
  4. Hecke algebra $\mathcal{H}_m$ and the Yang-Baxter equation
  5. Characters of $\mathcal{H}_m$
  6. Quantum immanants and Schur-Weyl-Jimbo duality
  7. Quantum immanants
  8. Gelfand-Tsetlin basis of $U_q(\mathfrak{gl}_n)$
  9. Schur-Weyl-Jimbo duality
  10. The proof of Theorem \ref{['q-Kostant-thm']}
  11. Quantum Littlewood correspondence I and II
  12. $q$-Goulden-Jackson identities and quantum Littlewood correspondence III
  13. The Bethe subalgebra of $A_q(\mathrm{Mat}_n)$
  14. $q$-Goulden-Jackson identities and Quantum Littlewood correspondence III

Key Result

Proposition 2.1

Let $\lambda=(\lambda_1,\dots,\lambda_l)\vdash m$ and $\mathcal{T}$ is a standard Young tableau of shape $\lambda$, then

Theorems & Definitions (40)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 30 more