Quotients of graph operators by symmetry representations

TL;DR

This paper introduces a simplified, generalized method for constructing quotient operators of graph operators that respect symmetry representations, applicable to both discrete and quantum graphs, enhancing spectral analysis and isospectrality studies.

Contribution

It presents a new, computationally simple basis for intertwiners to construct quotient operators that preserve structure, extending previous methods to quantum graphs and addressing open questions.

Findings

Provides a unified framework for quotient operators in graphs and quantum graphs.

Simplifies previous constructions and extends applicability.

Demonstrates effectiveness through numerous examples and applications.

Abstract

A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen representation and reduces the complexity of the spectral problem. Yet, such a quotient operator is not uniquely defined. Here we present a computationally simple way of choosing a special basis for the space of intertwiners, allowing us to construct a quotient that reflects the structure of the original operator. This quotient construction generalizes previous definitions for discrete graphs, which either dealt with restricted group actions or only with the trivial representation. We also extend the method to quantum graphs, which simplifies previous constructions within this context, answers an open question regarding self-adjointness and offers alternative viewpoints in terms of a scattering approach. Applications to isospectrality are discussed, together with numerous examples and comparisons with previous results.