Representations of shifted super Yangians and finite $W$-superalgebras of type A

TL;DR

This paper explores the representation theory of shifted super Yangians and finite W-superalgebras of type A, providing criteria for finite-dimensional modules, explicit character formulas, and insights into their centers.

Contribution

It introduces a finite-dimensionality criterion for irreducible modules, derives an explicit Gelfand-Tsetlin character formula, and shows the centers of related W-superalgebras are isomorphic.

Findings

Finite-dimensionality criterion for irreducible modules.

Explicit Gelfand-Tsetlin character formula for Verma modules.

Centers of finite W-superalgebras associated to even nilpotent elements are isomorphic.

Abstract

In this article, we study the representation theory of shifted super Yangians and finite $W$-superalgebras of type A. A criterion for the finite dimensionality of irreducible modules is obtained in the standard parity case. Furthermore, we provide an explicit Gelfand-Tsetlin character formula for Verma modules of finite $W$-superalgebras. As an application, we show that the centers of the finite $W$-superalgebras associated to any even nilpotent elements belonging to the same general linear Lie superalgebra are all isomorphic to the center of the universal enveloping superalgebra.