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Jacobi-Trudi identity and Drinfeld functor for super Yangian

Kang Lu, Evgeny Mukhin

TL;DR

This work analyzes the representation theory and integrable structure of the super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$ by constructing skew representations $L(\lambda/\mu)$ and establishing their $q$-character theory via a Jacobi–Trudi framework. It then connects these skew modules to degenerate affine Hecke algebras through the Drinfeld functor, yielding irreducibility criteria for tensor products and extended $T$-systems, all within the super setting. A central result is the factorization of the quantum Berezinian as $\mathfrak D_1\mathfrak D_2^{-1}$, where $\mathfrak D_1$ and $\mathfrak D_2$ are difference operators of orders $m$ and $n$ whose coefficients are transfer matrices associated to skew diagrams, up to a normalization by the $m\times n$ rectangle. The paper also develops Harish-Chandra maps and a rational form for the Berezinian, linking eigenvalues of transfer matrices to $q$-character divisibility and Bethe ansatz data, thereby providing a unified operator-level description of the integrals of motion and spectral data in the supersymmetric case.

Abstract

We show that the quantum Berezinian which gives a generating function of the integrals of motions of XXX spin chains associated to super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$ can be written as a ratio of two difference operators of orders $m$ and $n$ whose coefficients are ratios of transfer matrices corresponding to explicit skew Young diagrams. In the process, we develop several missing parts of the representation theory of $\mathrm{Y}(\mathfrak{gl}_{m|n})$ such as $q$-character theory, Jacobi-Trudi identity, Drinfeld functor, extended T-systems, Harish-Chandra map.

Jacobi-Trudi identity and Drinfeld functor for super Yangian

TL;DR

This work analyzes the representation theory and integrable structure of the super Yangian by constructing skew representations and establishing their -character theory via a Jacobi–Trudi framework. It then connects these skew modules to degenerate affine Hecke algebras through the Drinfeld functor, yielding irreducibility criteria for tensor products and extended -systems, all within the super setting. A central result is the factorization of the quantum Berezinian as , where and are difference operators of orders and whose coefficients are transfer matrices associated to skew diagrams, up to a normalization by the rectangle. The paper also develops Harish-Chandra maps and a rational form for the Berezinian, linking eigenvalues of transfer matrices to -character divisibility and Bethe ansatz data, thereby providing a unified operator-level description of the integrals of motion and spectral data in the supersymmetric case.

Abstract

We show that the quantum Berezinian which gives a generating function of the integrals of motions of XXX spin chains associated to super Yangian can be written as a ratio of two difference operators of orders and whose coefficients are ratios of transfer matrices corresponding to explicit skew Young diagrams. In the process, we develop several missing parts of the representation theory of such as -character theory, Jacobi-Trudi identity, Drinfeld functor, extended T-systems, Harish-Chandra map.

Paper Structure

This paper contains 27 sections, 52 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$
  3. Lie superalgebra $\mathfrak{gl}_{m|n}$
  4. Hook partitions, skew Young diagrams, and polynomial modules
  5. Super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$
  6. Gauss decomposition
  7. Highest and lowest $\ell$-weight representations
  8. Evaluation maps
  9. Category $\mathcal{C}$ and $q$-character map
  10. Skew representations and Jacobi-Trudi identity
  11. Skew representations
  12. $q$-characters of skew representations
  13. Divisibility of $q$-characters
  14. Jacobi-Trudi type identity
  15. Drinfeld functor and skew representations
  16. ...and 12 more sections

Key Result

Theorem 2.3

The $\mathfrak S_l$-action and $\mathfrak{gl}_{m|n}$-action on $V^{\otimes l}$ commute. Moreover, as a $\mathrm{U}(\mathfrak{gl}_{m|n})\otimes \mathbb{C}[\mathfrak S_l]$-module, we have

Figures (1)

  • Figure 1: Skew Young diagrams corresponding to the coefficients of the difference operators.

Theorems & Definitions (103)

  • Example 2.1
  • Example 2.2
  • Theorem 2.3: Schur-Sergeev duality Ser85
  • Theorem 2.4: Gow07
  • Lemma 2.5: Gow07
  • Theorem 2.6: Z96
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 93 more