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On the Mathieu Conjecture for $SU(N)$ and $SO(N)$

Kevin Zwart

Abstract

Building on work of M. Müger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for $SU(N)$ to a simpler conjecture in purely abelian terms. We sketch a similar reduction for $SO(N)$. The proofs rely on Euler-style parametrizations of these groups, which we discuss including proofs.

On the Mathieu Conjecture for $SU(N)$ and $SO(N)$

Abstract

Building on work of M. Müger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for to a simpler conjecture in purely abelian terms. We sketch a similar reduction for . The proofs rely on Euler-style parametrizations of these groups, which we discuss including proofs.
Paper Structure (4 sections, 9 theorems, 151 equations)

This paper contains 4 sections, 9 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. The case of $SU(N)$
  3. The case of $SO(N)$
  4. Appendix

Key Result

lemma 2.2

Let $N\geq 2$. Define inductively the mapping $F_N:([0,\pi]\times[0,2\pi]^{N-2})\times([0,\pi]\times[0,2\pi]^{N-3})\times\cdots\times([0,\pi]\times [0,2\pi])\times[0,\pi]\times \left[0,\frac{\pi}{2}\right]^{\frac{N(N-1)}{2}}\times[0,2\pi]\times\cdots\times\left[0,\frac{2\pi}{N-1}\right]\rightarrow S where $A(k)(x,y):=e^{\lambda_{3}x}e^{\lambda_{(k-1)^2+1}y}$, and $\psi_j\in\left[0,\frac{\pi}{2}\ri

Theorems & Definitions (27)

  • conjecture 1.1: Jacobi Conjecture
  • definition 2.1
  • conjecture 2.2: The Mathieu Conjecture Mathieu
  • lemma 2.2: Generalized Euler Angles
  • remark 1
  • lemma 2.2
  • remark 2
  • lemma 2.3
  • lemma 2.4
  • proof
  • ...and 17 more