Representation Theory and Differential Equations

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 \(\mathbb Z/3\mathbb Z\) $\mathbb{Z}/3\mathbb{Z}$ , \(\displaystyle \Delta _3=\dfrac{\partial ^3}{\partial x^3}+\dfrac{\partial ^3}{\partial y^3}+\dfrac{\partial ^3}{\partial z^3}-3\dfrac{\partial ^3}{\partial x\partial y\partial z}\) ${\displaystyle {\mathrm{\Delta }}_3={\displaystyle \frac{{\partial }^3}{\partial x^3}}+{\displaystyle \frac{{\partial }^3}{\partial y^3}}+{\displaystyle \frac{{\partial }^3}{\partial z^3}}-3{\displaystyle \frac{{\partial }^3}{\partial x\partial y\partial z}}}$ . 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.