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A note on odd reflections of super Yangian and Bethe ansatz

Kang Lu

TL;DR

The note reframes the odd reflections for the super Yangian $Y(\mathfrak{gl}_{m|n})$ using the Drinfeld-current presentation, deriving explicit weight-transition rules under parity switches and linking them to the fermionic reproduction procedure in XXX-type Bethe ansatz equations. It provides an algorithm to update $q$-characters under odd reflections and computes $q$-characters for skew $Y(\mathfrak{gl}_{m|n})$-modules for arbitrary parity sequences, showing independence of parity for these skew modules. The work bridges representation theory of super Yangians with integrable models, generalizing the reflection phenomena beyond tensor products of evaluation modules and clarifying the interplay with Bethe ansatz and Gelfand–Tsetlin structures.

Abstract

In a recent paper arXiv:2109.09462, Molev introduced analogues of the odd reflections for the super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$ and obtained a transition rule for the change of highest weights when the parity sequence is altered. In this note, we reproduce the results from a different point of view and discuss their relations with the fermionic reproduction procedure of the XXX-type Bethe ansatz equations introduced in arXiv:1811.11225. We give an algorithm that how the $q$-characters change under the odd reflections. We also take the chance to compute explicitly the $q$-characters of skew representations of $\mathrm{Y}(\mathfrak{gl}_{m|n})$ for arbitrary parity sequences.

A note on odd reflections of super Yangian and Bethe ansatz

TL;DR

The note reframes the odd reflections for the super Yangian using the Drinfeld-current presentation, deriving explicit weight-transition rules under parity switches and linking them to the fermionic reproduction procedure in XXX-type Bethe ansatz equations. It provides an algorithm to update -characters under odd reflections and computes -characters for skew -modules for arbitrary parity sequences, showing independence of parity for these skew modules. The work bridges representation theory of super Yangians with integrable models, generalizing the reflection phenomena beyond tensor products of evaluation modules and clarifying the interplay with Bethe ansatz and Gelfand–Tsetlin structures.

Abstract

In a recent paper arXiv:2109.09462, Molev introduced analogues of the odd reflections for the super Yangian and obtained a transition rule for the change of highest weights when the parity sequence is altered. In this note, we reproduce the results from a different point of view and discuss their relations with the fermionic reproduction procedure of the XXX-type Bethe ansatz equations introduced in arXiv:1811.11225. We give an algorithm that how the -characters change under the odd reflections. We also take the chance to compute explicitly the -characters of skew representations of for arbitrary parity sequences.
Paper Structure (21 sections, 17 theorems, 83 equations)

This paper contains 21 sections, 17 theorems, 83 equations.

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. Category $\mathcal{C}$ and $q$-character map
  9. More on super Yangians
  10. Quasi-determinant
  11. Homomorphisms between super Yangians
  12. Isomorphisms between super Yangians of different parity sequences
  13. Odd reflections in Drinfeld current presentation
  14. Odd reflections of super Yangian
  15. The case of $\mathfrak{gl}_{1|1}$
  16. ...and 6 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

Theorems & Definitions (36)

  • Example 2.1
  • Example 2.2
  • Theorem 2.3: Schur-Sergeev duality Ser:1985
  • Theorem 2.4: Zh:1996
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 3.1: Gow:2007P:2016
  • ...and 26 more