Jacobi-Trudi identity and Drinfeld functor for super Yangian

TL;DR

This paper links the quantum Berezinian of super Yangian spin chains to difference operators derived from transfer matrices, advancing the representation theory of $ ext{Y}( ext{gl}_{m|n})$ with new identities and functors.

Contribution

It develops key aspects of the representation theory of super Yangians, including the Jacobi-Trudi identity and Drinfeld functor, and relates the quantum Berezinian to difference operators.

Findings

Quantum Berezinian expressed as ratio of difference operators.

Established new identities in super Yangian representation theory.

Connected transfer matrices to skew Young diagrams.

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.