Weekly seminars

# Construction of Gelfand-Tsetlin modules for $gl(n)$

## par Vyacheslav Futorny (Univ. Sao Paolo)

Europe/Paris
Auditorium (Annecy-le-Vieux)

### Auditorium

#### Annecy-le-Vieux

9 chemin de Bellevue 74940 ANNECY LE VIEUX
Description
A classical paper of Gelfand and Tsetlin describes a basis of irreducible finite dimensional modules over the Lie algebra $gl(n)$. This is one of the most remarkable results of the representation theory of Lie algebras which initiated a development of the theory of Gelfand-Tsetlin modules. The Gelfand-Tsetlin modules form the largest subcategory of $gl(n)$-modules (in particular, of weight modules with respect to a fixed Cartan subalgebra) where there is some understanding of irreducible modules. The main remaining problem is how to construct explicitly these modules. We propose a new effective method of constructing explicitly Gelfand -Tsetlin modules for $gl(n)$ and obtain a large family of irreducible modules that have a basis consisting of Gelfand-Tsetlin tableaux and the action of the Lie algebra is given by the Gelfand-Tsetlin formulas. As an application of our construction we prove necessary and sufficient condition for the Gelfand and Graev's continuation construction to define a module which was conjectured by Lemire and Patera. The talk is based on recent joint results with Luis Enrique Ramirez and Jian Zhang.
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