A functional realization of the Gelfand-Tsetlin base

TL;DR

This paper constructs a new realization of finite-dimensional irreducible representations of rak{gl}_n using functions related to hypergeometric functions, connecting Gelfand-Tsetlin bases with hypergeometric systems.

Contribution

It introduces a novel hypergeometric-based realization of Gelfand-Tsetlin bases, linking representation theory with hypergeometric functions and differential equations.

Findings

Functions corresponding to Gelfand-Tsetlin diagrams are expressed as linear combinations of hypergeometric-type functions.

These functions satisfy a system of PDEs derived from the Gelfand-Kapranov-Zelevinsky system.

Coefficients in the linear combinations are hypergeometric constants, evaluated as hypergeometric functions.

Abstract

In the paper we consider a realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $GL_n$. It is proved that functions corresponding to Gelfand-Tsetlin diagrams are linear combinations of some new functions of hypergeometric type which are closely related to $A$-hypergeometric functions. These new functions are solution of a system of partial differential equations which one obtains from the Gelfand-Kapranov-Zelevinsky by an "antisymmetrization". The coefficients in the constructed linear combination are hypergeometric constants i.e. they are values of some hypergeometric functions when instead of all arguments ones are substituted.