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HCIZ integral formula as unitarity of a canonical map between reproducing kernel spaces

Martin Miglioli

Abstract

In this article we prove that the Harish-Chandra-Itzykson-Zuber (HCIZ) integral formula is equivalent to the unitarity of a canonical map between invariant subspaces of Segal-Bargmann spaces. As a consequence, we provide two new proofs of the HCIZ integral formula and alternative proofs of related results.

HCIZ integral formula as unitarity of a canonical map between reproducing kernel spaces

Abstract

In this article we prove that the Harish-Chandra-Itzykson-Zuber (HCIZ) integral formula is equivalent to the unitarity of a canonical map between invariant subspaces of Segal-Bargmann spaces. As a consequence, we provide two new proofs of the HCIZ integral formula and alternative proofs of related results.
Paper Structure (4 sections, 10 theorems, 88 equations)

This paper contains 4 sections, 10 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Invariant subspaces of Segal-Bargmann spaces
  3. HCIZ integral formula as unitarity of a canonical map
  4. Differentiation of polynomials and orthonormal bases

Key Result

Proposition 2.2

The reproducing kernel on $\mathop{\mathrm{\mathcal{F}}}\nolimits(\mathop{\mathrm{\mathbb{C}}}\nolimits^{n\times n})^U$ is given by for $z,a\in \mathop{\mathrm{\mathbb{C}}}\nolimits^{n\times n}$.

Theorems & Definitions (23)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 13 more