The Gelfand-Tsetlin basis for irreducible representations of an infinite-dimensional general linear group

2024/07/16 by Movchan, Evgenii
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2407.12199

Abstract

We consider the problem of constructing a Gelfand--Tsetlin basis in irreducible representations of an infinite-dimensional general linear group. For a finite-dimensional irreducible representation of a general linear group, all elements of the Gelfand--Tsetlin basis are parameterized by Gelfand--Tsetlin schemes. We extend this definition to infinite Gelfand--Tsetlin schemes, which in turn parameterize elements of the Gelfand--Tsetlin basis of an irreducible representation of an infinite-dimensional complete linear group. Using the properties of colimits of representations with the highest weight, we present an explicit form of the Gelfand--Tsetlin basis.

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