Spectral decomposition of hypergraph automorphism compatible matrices

TL;DR

This paper investigates how the spectral properties of matrices associated with hypergraphs relate to their automorphisms, revealing that certain symmetries allow the spectrum to be decomposed into smaller, more manageable parts.

Contribution

It introduces the concept of f-compatible matrices for hypergraph automorphisms and demonstrates spectral decomposition for rotations and unit-automorphisms, expanding understanding of hypergraph symmetries.

Findings

Spectra of f-compatible matrices decompose into spectra of smaller matrices.

Rotation automorphisms enable spectral decomposition of associated matrices.

Unit-automorphisms also induce spectral decompositions for compatible matrices.

Abstract

This study explores the relationship between hypergraph automorphisms and the spectral properties of matrices associated with hypergraphs. For an automorphism $f$, an \( f \)-compatible matrices capture aspects of the symmetry, represented by \( f \), within the hypergraph. First, we explore rotation, a specific kind of automorphism and find that the spectrum of any matrix compatible with a rotation can be decomposed into the spectra of smaller matrices associated with that rotation. We show that the spectrum of any \(f\)-compatible matrix can be decomposed into the spectra of smaller matrices associated with the component rotations comprising \( f \). Further, we study a hypergraph symmetry termed unit-automorphism, which induces bijections on the hyperedges, though not necessarily on the vertex set. We show that unit automorphisms also lead to the spectral decomposition of compatible matrices.