Computing Hypermatrix Spectra with the Poisson Product Formula

TL;DR

This paper calculates the spectrum of the all-ones hypermatrix and sunflower hypergraphs using the Poisson product formula, providing detailed eigenvalue multiplicities and spectral distribution insights.

Contribution

It introduces a novel application of the Poisson product formula to compute hypermatrix spectra, including multiplicities and spectral distribution for specific hypergraph classes.

Findings

Complete eigenvalue multiplicity descriptions

Distributional analysis of the spectrum in the complex plane

Spectral characterization of sunflower hypergraphs

Abstract

We compute the spectrum of the "all ones" hypermatrix using the Poisson product formula. This computation includes a complete description of the eigenvalues' multiplicities, a seemingly elusive aspect of the spectral theory of tensors. We also give a general distributional picture of the spectrum as a point-set in the complex plane, and use our techniques to analyze the spectrum of "sunflower hypergraphs", a class that has played a prominent role in extremal hypergraph theory.