Gelfand-Tsetlin Theory for Rational Galois Algebras

Futorny, Vyacheslav; Grantcharov, Dimitar; Ramírez, Luis Enrique; Zadunaisky, Pablo M.

doi:10.1007/s11856-020-2048-2

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Gelfand-Tsetlin Theory for Rational Galois Algebras

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Zadunaisky, Pablo M.
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s11856-020-2048-2
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Citation

Futorny, V., Grantcharov, D., Ramírez, L. E., & Zadunaisky, P. M. (2020). Gelfand-Tsetlin Theory for Rational Galois Algebras. Israel Journal of Mathematics, 239, 99-128. doi:10.1007/s11856-020-2048-2.

Cite as: https://hdl.handle.net/21.11116/0000-0007-8592-D

Abstract

In the present paper we study Gelfand-Tsetlin modules defined in terms of BGG
differential operators. The structure of these modules is described with the aid of the Postnikov-Stanley polynomials introduced in [PS09]. These polynomials are used to identify the action of the Gelfand-Tsetlin subalgebra on the BGG operators. We also provide explicit bases of the corresponding Gelfand-Tsetlin modules and prove a simplicity criterion for these modules. The results hold for modules defined over standard Galois orders of type $A$ - a large class of rings that include the universal enveloping algebra of
$\mathfrak{gl} (n)$ and the finite $W$-algebras of type $A$.