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On the Mathieu Conjecture for $Sp(N)$ and $G_2$

Kevin Zwart

TL;DR

The paper extends the reduction of Mathieu's conjecture to the nonabelian compact groups Sp$(N)$ and $G_2$ by constructing a generalized Euler angles (a KAK-type) decomposition and an explicit Haar-measure parametrization. Finite-type functions on these groups are expressed in coordinates that separate a $(K/M)$ factor, a toral abelian part $A$, and a $K$-component, yielding an abelian-like integral with a weight. Central to the approach are the notions of $\frac{1}{N}$-admissible functions and their spectra, plus a convex-hull conjecture akin to Müger–Zwart's abelian reductions; these reduce the nonabelian Mathieu conjecture to questions about the convex geometry of the transformed spectra. Assuming the proposed convex-hull conjectures for Sp$(N)$ and $G_2$, the Mathieu conjecture holds for these groups, with proofs leveraging Haar-invariance arguments to derive linear relations among exponential exponents and force the abelian integrals to dictate the conclusion. The work thus provides a pathway to extending Mathieu’s conjecture to broader Lie groups via explicit homogeneous-space parameterizations and abelian reductions.

Abstract

As a direct continuation of K. Zwart, arXiv:2304.02648, which is built on the work of M. Müger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for $Sp(N)$ and $G_2$ to a conjecture involving functions over $\mathbb{R}^n\times (S^1)^m$ with $n,m\in\mathbb{N}_0$. The proofs rely on Euler-style parametrizations of these groups, a specific version of the $KAK$ decomposition, which we discuss and prove.

On the Mathieu Conjecture for $Sp(N)$ and $G_2$

TL;DR

The paper extends the reduction of Mathieu's conjecture to the nonabelian compact groups Sp and by constructing a generalized Euler angles (a KAK-type) decomposition and an explicit Haar-measure parametrization. Finite-type functions on these groups are expressed in coordinates that separate a factor, a toral abelian part , and a -component, yielding an abelian-like integral with a weight. Central to the approach are the notions of -admissible functions and their spectra, plus a convex-hull conjecture akin to Müger–Zwart's abelian reductions; these reduce the nonabelian Mathieu conjecture to questions about the convex geometry of the transformed spectra. Assuming the proposed convex-hull conjectures for Sp and , the Mathieu conjecture holds for these groups, with proofs leveraging Haar-invariance arguments to derive linear relations among exponential exponents and force the abelian integrals to dictate the conclusion. The work thus provides a pathway to extending Mathieu’s conjecture to broader Lie groups via explicit homogeneous-space parameterizations and abelian reductions.

Abstract

As a direct continuation of K. Zwart, arXiv:2304.02648, which is built on the work of M. Müger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for and to a conjecture involving functions over with . The proofs rely on Euler-style parametrizations of these groups, a specific version of the decomposition, which we discuss and prove.
Paper Structure (9 sections, 13 theorems, 173 equations)

This paper contains 9 sections, 13 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. The case of $Sp(N)$
  3. The Euler angles decomposition of $Sp(N)$
  4. The Mathieu Conjecture on $Sp(N)$
  5. The case of $G_2$
  6. The Euler angle decomposition of $G_2$
  7. The Mathieu Conjecture on $G_2$
  8. Proof of Theorem \ref{['thm:KAK_decomp']}
  9. Some results regarding $G_2$

Key Result

theorem 2.2

[The Euler Angles Theorem] Let $G$ be a simply connected compact Lie group with finite center, and let $\mathfrak{g}$ be its Lie algebra. Let $\theta:\mathfrak{g}\rightarrow\mathfrak{g}$ be an involutive automorphism. Let $\mathfrak{k}, \mathfrak{p}$ be the $+1$ and $-1$ eigenspace of $\theta$, resp is surjective and a diffeomorphism up to a measure zero set if we replace $\mathcal{A}$ by $\mathrm

Theorems & Definitions (32)

  • conjecture 1.1
  • definition 2.1
  • conjecture 2.2: The Mathieu Conjecture Mathieu
  • theorem 2.2
  • definition 2.3
  • lemma 2.3: Generalized Euler Angles $SU(N)$, Zwart
  • corollary 2.4: Generalized Euler Angles $U(N)$
  • proof
  • corollary 2.5: Parametrization $U(N)/\mathbb{Z}_2^N$.
  • lemma 2.6: Euler Angles Parametrization of $Sp(N)$
  • ...and 22 more