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The Gelfand-Tsetlin type base for the algebra $\mathfrak{g}_2$

Dmitry Artamonov

TL;DR

The paper develops a Gelfand-Tsetlin-type framework for the exceptional Lie algebra $\mathfrak{g}_2$ by constructing the representation space as polynomial solutions to a hypergeometric-type system (A-GKZ) built on a Gelfand-Tsetlin lattice adapted from $\mathfrak{gl}_8$ and refined to $\mathfrak{g}_2$. This A-GKZ realization yields an explicit, invariant inner product and a canonical embedding of all finite-dimensional irreducibles with multiplicity one; GT-type basis vectors arise from orthogonalizing the natural A-GKZ solutions. The approach integrates GKZ hypergeometric theory with octonionic invariants and determinant relations, and it provides concrete formulations for the action of $\mathfrak{g}_2$ and its restriction to $\mathfrak{sl}_3$, allowing a GT-like decomposition along the chain $\mathfrak{g}_2 \supset \mathfrak{sl}_3$. These results illuminate deep connections between hypergeometric systems, GT lattices, and exceptional Lie algebra representation theory, with potential applications to tensor product decompositions and explicit computation of intertwiners. The framework is constructed to extend GT-type methodologies from classical series to the exceptional case of $\mathfrak{g}_2$.

Abstract

The paper presents a construction of finite-dimensional irreducible representations of the Lie algebra $\mathfrak{g}_2$. The representation space is constructed as the space of solutions to a certain system of partial differential equations of hypergeometric type, which is closely related to the Gelfand-Kapranov-Zelevinsky systems. This connection allows for the construction of a basis in the representation. The orthogonalization of the constructed basis with respect to the invariant scalar product turns out to be a Gelfand-Tsetlin-type basis for the chain of subalgebras $\mathfrak{g}_2 \supset \mathfrak{sl}_3$.

The Gelfand-Tsetlin type base for the algebra $\mathfrak{g}_2$

TL;DR

The paper develops a Gelfand-Tsetlin-type framework for the exceptional Lie algebra by constructing the representation space as polynomial solutions to a hypergeometric-type system (A-GKZ) built on a Gelfand-Tsetlin lattice adapted from and refined to . This A-GKZ realization yields an explicit, invariant inner product and a canonical embedding of all finite-dimensional irreducibles with multiplicity one; GT-type basis vectors arise from orthogonalizing the natural A-GKZ solutions. The approach integrates GKZ hypergeometric theory with octonionic invariants and determinant relations, and it provides concrete formulations for the action of and its restriction to , allowing a GT-like decomposition along the chain . These results illuminate deep connections between hypergeometric systems, GT lattices, and exceptional Lie algebra representation theory, with potential applications to tensor product decompositions and explicit computation of intertwiners. The framework is constructed to extend GT-type methodologies from classical series to the exceptional case of .

Abstract

The paper presents a construction of finite-dimensional irreducible representations of the Lie algebra . The representation space is constructed as the space of solutions to a certain system of partial differential equations of hypergeometric type, which is closely related to the Gelfand-Kapranov-Zelevinsky systems. This connection allows for the construction of a basis in the representation. The orthogonalization of the constructed basis with respect to the invariant scalar product turns out to be a Gelfand-Tsetlin-type basis for the chain of subalgebras .

Paper Structure

This paper contains 31 sections, 21 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Construction of representations
  3. The structure of the paper
  4. The Lie group $G_2$ and the Lie algebra $\mathfrak{g}_2$. A functional representation of representations
  5. The Lie algebra $\mathfrak{g}_2$
  6. The functional realzation of representations
  7. The Lie group $G_2$. Relations between determinants
  8. The relations coming from the embedding $G_2\subset GL(8)$
  9. The relations coming from the embedding $G_2\subset SO(8)$
  10. The relations specific for $G_2$
  11. The basis is ideally the ratio between the determinants
  12. The A-GKZ system and the A-GKZ model of representations for the algebra $\mathfrak{gl_8}$. The Gelfand-Tsetlin type basis.
  13. $\Gamma$-series and the GKZ system in the general case
  14. The Gelfand-Tsetlin lattice associated with $\mathfrak{gl}_8$
  15. The GKZ and A-GKZ systems
  16. ...and 16 more sections

Key Result

Lemma 1

One has Here

Theorems & Definitions (32)

  • Lemma 1: The Jacobi relations for determinants
  • Lemma 2: a2
  • Theorem 1
  • Definition 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Definition 2
  • Definition 3
  • ...and 22 more