Singular Gelfand-Tsetlin modules of gl(n)

Author(s)

Futorny, Vyacheslav ^[1] ; Grantcharov, Dimitar ^[2] ; Ramirez, Luis Enrique ^[1]

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Sao Paulo, SP - Brazil

^[2] Univ Texas Arlington, Arlington, TX 76019 - USA

• Document type: Journal article
• Source: ADVANCES IN MATHEMATICS; v. 290, p. 453-482, FEB 26 2016.
• Web of Science Citations: 15

Abstract

The classical Gelfand-Tsetlin formulas provide a basis in terms of tableaux and an explicit action of the generators of gl(n) for every irreducible finite-dimensional gl{[}(n)-module. These formulas can be used to define a gl(n)-module structure on some infinite-dimensional modules - the so-called generic Gelfand-Tsetlin modules. The generic Gelfand-Tsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic Gelfand-Tsetlin module containing this tableau as a basis element. In this paper we initiate the systematic study of a large class of non-generic Gelfand-Tsetlin modules - the class of 1-singular Gelfand-Tsetlin modules. An explicit tableaux realization and the action of gl(n) on these modules is provided using a new construction which we call derivative tableaux. Our construction of 1-singular modules provides a large family of new irreducible Gelfand-Tsetlin modules of gl(n), and is a part of the classification of all such irreducible modules for n = 3. (C) 2015 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 11/21621-8 - Representations of Lie (super)algebras of vector fields
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• FAPESP's process: 14/09310-5 - Algebraic structures and their representations
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 12/23450-9 - Gelfand-Tsetlin modules for Lie algebras
• Grantee: Luis Enrique Ramírez
• Support Opportunities: Scholarships in Brazil - Post-Doctoral