New singular Gelfand-Tsetlin $\mathfrak{gl} (n)$-modules of index 2

TL;DR

This paper constructs new classes of singular Gelfand-Tsetlin modules of index 2 for rak{gl}(n), providing explicit bases and actions, thus expanding the understanding of these modules and their irreducibility.

Contribution

It introduces a general construction of singular Gelfand-Tsetlin modules of index 2 for any singular character, with explicit bases and generator actions.

Findings

New families of irreducible Gelfand-Tsetlin modules

Explicit bases of derivative tableaux provided

Tables bases for some simple Verma modules

Abstract

Singular Gelfand-Tsetlin modules of index 2 are modules whose tableaux bases may have singular pairs but no singular triples of entries on each row. In this paper we construct singular Gelfand-Tsetlin modules for arbitrary singular character of index 2. Explicit bases of derivative tableaux and the action of the generators of $\mathfrak{gl}(n)$ are given for these modules. Our construction leads to new families of irreducible Gelfand-Tsetlin modules and also provides tableaux bases for some simple Verma modules.