Representation Theory and Differential Equations

TL;DR

This paper explores the connections between representation theory, group-determinants, and differential equations, highlighting the spectral properties of operators like the Humbert operator and their relation to Fourier analysis.

Contribution

It introduces new links between spectral theory of Frobenius determinants and finite Fourier transforms, extending classical differential operators in the context of group representations.

Findings

Spectral theory of Frobenius determinants relates to finite Fourier transforms.

Humbert operator extends the Laplacian in dimension 2.

New insights into differential equations from group-determinants and representation theory.

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.