A note on odd reflections of super Yangian and Bethe ansatz

TL;DR

This paper explores the effects of odd reflections on super Yangian representations, providing new algorithms for $q$-character transformations and explicit computations for skew representations, linking to Bethe ansatz methods.

Contribution

It offers a novel perspective on odd reflections in super Yangians, introduces an algorithm for $q$-character changes, and computes explicit $q$-characters for skew representations.

Findings

Derived an algorithm for $q$-character transformations under odd reflections.

Explicitly computed $q$-characters of skew representations for arbitrary parity sequences.

Connected odd reflections with the fermionic reproduction procedure of Bethe ansatz equations.

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.