Relation modules for finite W-algebras and tensor products of highest weight evaluation modules for Yangians of type A

TL;DR

This paper constructs Gelfand-Tsetlin modules for finite W-algebras of type A, characterizes their irreducibility, and extends irreducibility results for tensor products of Yangian modules with generic highest weights.

Contribution

It explicitly constructs a large family of irreducible Gelfand-Tsetlin modules for finite W-algebras and extends irreducibility criteria for Yangian tensor products.

Findings

Constructed Gelfand-Tsetlin modules with admissible relations.

Proved irreducibility of these modules.

Extended irreducibility results to tensor products of Yangian modules.

Abstract

We construct explicitly a large family of Gelfand-Tsetlin modules for an arbitrary finite W-algebra of type A and establish their irreducibility. A basis of these modules is formed by the Gelfand-Tsetlin tableaux whose entries satisfy certain admissible sets of relations. Characterization and an effective method of constructing such admissible relations are given. In the case of the Yangian of gl_n we prove the sufficient condition for the irreducibility of the tensor product of two highest weight relation modules and establish irreducibility of any number of highest weight relation modules with generic highest weights. This extends the results of Molev to infinite dimensional highest modules.