A functional realization of the Gelfand-Tsetlin base

TL;DR

This work builds a functional realization of finite-dimensional $\mathfrak{gl}_n$-modules in the space of functions on $GL_n$ by encoding GT diagrams as linear combinations of new hypergeometric-type functions related to $A$-hypergeometric theory. Central to the construction is the $A$-GKZ framework and its irreducible solutions $F_{\gamma}$, which are tied to a GT-structured lattice, with the antisymmetrized GKZ system providing a PDE backbone. The GKZ base $\mathcal{F}_{\gamma}(a)$ is shown to form a representation basis that connects to the classical GT base via a triangular (and in part lower-triangular) change of basis, with explicit coefficient formulas. The coefficients in the GT-to-GKZ expansions are hypergeometric constants, expressible as Horn-function values, linking combinatorial GT data to analytic hypergeometric objects and enabling explicit computation of GT-related quantities such as Clebsch–Gordan coefficients within this analytic framework.

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.

Theorems & Definitions (35)

• Remark 1
• Theorem 1
• Definition 1
• Definition 2
• Proposition 1
• Proof 1
• Definition 3
• Definition 4
• Lemma 4.1
• Definition 5