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Representation Theory and Differential Equations

Ahmed Sebbar, Oumar Wone

Abstract

We study the geometry and partial differential equations arising from the consideration of group-determinants, and representation theory. The simplest and most striking such example is undoubtedly that of the Humbert operator, associated with the cyclic group Z/3Z. This operator appears as a natural extension of the Laplacian in dimension 2. Another originality of our work is to show that the spectral theory of operators associated with Frobenius determinants is closely linked to finite Fourier transform theory.

Representation Theory and Differential Equations

Abstract

We study the geometry and partial differential equations arising from the consideration of group-determinants, and representation theory. The simplest and most striking such example is undoubtedly that of the Humbert operator, associated with the cyclic group Z/3Z. This operator appears as a natural extension of the Laplacian in dimension 2. Another originality of our work is to show that the spectral theory of operators associated with Frobenius determinants is closely linked to finite Fourier transform theory.

Paper Structure

This paper contains 9 sections, 10 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Partial differential equations defined by group-determinants
  3. "The kernels" of the PDEs arising from Frobenius determinants
  4. Eigenvalues problem
  5. Frobenius Groups, Almost-Frobenius structures and toric structures
  6. Frobenius Groups
  7. Almost-Frobenius structures
  8. Frobenius determinants and toric varieties
  9. Data Availability

Key Result

Theorem \oldthetheorem

For $G$ a finite abelian group one has where $\widehat{G}$ is the character group of $G$, i.e. the group of homomorphisms from $G\to \mathbb C^\times$.

Figures (1)

  • Figure 1: : A rotated Hexenhut: $z(x^2+y^2)=1$.

Theorems & Definitions (24)

  • Theorem \oldthetheorem: Dedekind, Frobenius, conrad1998
  • Example \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem: Frobenius
  • proof
  • Theorem \oldthetheorem
  • Example \oldthetheorem
  • ...and 14 more