Title: Quotients of finite-dimensional operators by symmetry representations,
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 problem,
however it 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.