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Gelfand–Tsetlin varieties for ${𝔤 𝔩}_{n}$

Germán Benitez

Department of Mathematics, Federal University of Amazonas, Manaus AM Brazil & Institute of Mathematics and Statistics, University of Sao Paulo, Sao Paulo SP, Brazil

E-mail Address: gabm@ufam.edu.br

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https://doi.org/10.1142/S0218196720500502Cited by:2 (Source: Crossref)

Abstract

Sergei Ovsienko proved that the Gelfand–Tsetlin variety for ${𝔤 𝔩}_{n}$ is equidimensional and the dimension of all irreducible components equals $n (n - 1) / 2$ . This implies in particular the equidimensionality of the nilfiber of the (partial) Kostant–Wallach map. We generalize this result for the $k$ -partial Kostant–Wallach map and prove that all its fibers are equidimensional of dimension $n^{2} - (k + 1) n + k (k + 1) / 2$ . Also, we study certain subvarieties of the Gelfand–Tsetlin variety and show their equidimensionality which gives a new proof of Ovsienko’s theorem for ${𝔤 𝔩}_{2}, {𝔤 𝔩}_{3}$ and ${𝔤 𝔩}_{4}$ .

Communicated by I. Shestakov

Keywords:

AMSC: 17B35, 14M10, 16W70

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