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Weyl group action on Radon hypergeometric function and its symmetry

Hironobu Kimura

TL;DR

This work establishes a Weyl-group–type symmetry for Radon hypergeometric functions $F_{\lambda}$ associated with partitions $\lambda$ of weight $n$ on Grassmannians, via the normalizer $N_G(H_{\lambda})$ and its quotient $W_{\lambda}=N_G(H_{\lambda})/H_{\lambda}$. It derives the precise structure of $W_{\lambda}$ as a semi-direct product $\bigl(\prod_i W_r(n_i)^{p_i}\bigr) \rtimes \mathcal{P}$, and shows how its continuous and finite parts act on the Radon HGF, yielding explicit transformation formulas for Hermitian matrix integral analogues of Gauss HGF and the Kummer confluent HGF. The paper then applies these symmetries to concrete examples for weight-3 and weight-4 partitions, obtaining matrix analogues of Beta/Gamma and Gauss/Kummer HGFs, plus transformation identities that mirror the classical 24 Gauss HGF solutions. Overall, the results unify classical HGF transformation theory within the Radon HGF framework, clarifying how the Weyl-group–like symmetry reduces parameter freedom and drives transformation formulas for matrix-integral HGFs.

Abstract

For positive integers $r,n,N:=rn$, we consider the Radon hypergeometric function (Radon HGF) associated with a partition $λ$ of $n$ defined on the Grassmannian $Gr(m,N)$ for $r<m<N$, which is obtained as the Radon transform of a character of the group $H_λ\subset G:=GL(N)$. We study its symmetry described by the Weyl group analogue $N_{G}(H_λ)/H_λ$. We consider the Hermitian matrix integral analogue of the Gauss HGF and its confluent family, which are understood as the Radon HGF on $Gr(2r,4r)$ for partitions $λ$ of $4$, we apply the result of symmetry to these particular cases and derive a transformation formula for the Gauss analogue which is known as a part of "24 solutions of Kummer" for the classical Gauss HGF. We derive a similar transformation formula for the analogue Kummer's confluent HGF.

Weyl group action on Radon hypergeometric function and its symmetry

TL;DR

This work establishes a Weyl-group–type symmetry for Radon hypergeometric functions associated with partitions of weight on Grassmannians, via the normalizer and its quotient . It derives the precise structure of as a semi-direct product , and shows how its continuous and finite parts act on the Radon HGF, yielding explicit transformation formulas for Hermitian matrix integral analogues of Gauss HGF and the Kummer confluent HGF. The paper then applies these symmetries to concrete examples for weight-3 and weight-4 partitions, obtaining matrix analogues of Beta/Gamma and Gauss/Kummer HGFs, plus transformation identities that mirror the classical 24 Gauss HGF solutions. Overall, the results unify classical HGF transformation theory within the Radon HGF framework, clarifying how the Weyl-group–like symmetry reduces parameter freedom and drives transformation formulas for matrix-integral HGFs.

Abstract

For positive integers , we consider the Radon hypergeometric function (Radon HGF) associated with a partition of defined on the Grassmannian for , which is obtained as the Radon transform of a character of the group . We study its symmetry described by the Weyl group analogue . We consider the Hermitian matrix integral analogue of the Gauss HGF and its confluent family, which are understood as the Radon HGF on for partitions of , we apply the result of symmetry to these particular cases and derive a transformation formula for the Gauss analogue which is known as a part of "24 solutions of Kummer" for the classical Gauss HGF. We derive a similar transformation formula for the analogue Kummer's confluent HGF.
Paper Structure (14 sections, 34 theorems, 213 equations)

This paper contains 14 sections, 34 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Radon HGF
  3. Jordan group
  4. Character of Jordan group
  5. Definition of HGF of type $\lambda$
  6. Weyl group analogue
  7. Statement of the result
  8. Proof of Theorem \ref{['thm:weyl-main-1']}
  9. Action of Weyl group on Radon HGF
  10. Examples
  11. Analogues of the beta and gamma functions
  12. Analogues of the Gauss HGF family
  13. Transformation formula for $\lambda=(1,1,1,1)$
  14. Transformation formula for Kummer's analogue

Key Result

Lemma 2.1

We have a group isomorphism defined by the correspondence $\mathrm{GL}(r)\ltimes J_{r}^{\circ}(p)\ni(g,h)\mapsto g\cdot h=\sum_{0\leq i<p}(gh_{i})\otimes\Lambda^{i}\in J_{r}(p)$, where the semi-direct product is defined by the action of $\mathrm{GL}(r)$ on $J_{r}^{\circ}(p)$: $h\mapsto g^{-1}hg=\sum_{i}(g^{-1}h_{i}g)\otimes\La

Theorems & Definitions (58)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Example
  • Definition 2.8
  • Proposition 2.9
  • ...and 48 more