This paper uncovers and exploits a link between a central object in harmonic
analysis, the so-called Schur functions, and the very hot topic of symmetry
protected topological phases of quantum matter. This connection is found in the
setting of quantum walks, i.e. quantum analogs of classical random walks. We
prove that topological indices classifying symmetry protected topological
phases of quantum walks are encoded by matrix Schur functions built out of the
walk. This main result of the paper reduces the calculation of these
topological indices to a linear algebra problem: calculating symmetry indices
of finite-dimensional unitaries obtained by evaluating such matrix Schur
functions at the symmetry protected points ±1. The Schur representation
fully covers the complete set of symmetry indices for 1D quantum walks with a
group of symmetries realizing any of the symmetry types of the tenfold way. The
main advantage of the Schur approach is its validity in the absence of
translation invariance, which allows us to go beyond standard Fourier methods,
leading to the complete classification of non-translation invariant phases for
typical examples.Comment: 33 pages, 1 figur