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Weekly seminars

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

by Vyacheslav Futorny (Univ. Sao Paolo)

jeudi 22 juin 2017 de à (Europe/Paris)
at Annecy-le-Vieux ( Auditorium )
9 chemin de Bellevue 74940 ANNECY LE VIEUX
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.