Mathematical Research Letters

Volume 28 (2021)

Number 5

Gelfand-Tsetlin modules for $\mathfrak{gl}(m \vert n)$

Pages: 1379 – 1418

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n5.a5

Authors

Vyacheslav Futorny (Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil; and International Center for Mathematics, SUSTech, Shenzhen, China)

Vera Serganova (Department of Mathematics, University of California, Berkeley, Calif., U.S.A.)

Jian Zhang (School of Mathematics and Statistics, Central China Normal University, Wuhan, China)

Abstract

We address the problem of classifying irreducible Gelfand–Tsetlin modules for $\mathfrak{gl}(m|n)$ and show that it reduces to the classification of Gelfand–Tsetlin modules for the even part. We also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand–Tsetlin modules which includes all essentially typical and covariant tensor finite dimensional modules. In the quasi typical case new irreducible representations are infinite dimensional $\mathfrak{gl}(m|n)$-modules which are isomorphic to the parabolically induced (Kac) modules.

Received 15 February 2020

Accepted 3 May 2020

Published 16 August 2022