  For quantum search via the continuous-time quantum walk, the evolution of the
whole system is usually limited in a small subspace. In this paper, we discuss
how the symmetries of the graphs are related to the existence of such an
invariant subspace, which also suggests a dimensionality reduction method based
on group representation theory. We observe that in the one-dimensional subspace
spanned by each desired basis state which assembles the identically evolving
original basis states, we always get a trivial representation of the symmetry
group. So we could find the desired basis by exploiting the projection operator
of the trivial representation. Besides being technical guidance in this type of
problem, this discussion also suggests that all the symmetries are used up in
the invariant subspace and the asymmetric part of the Hamiltonian is very
important for the purpose of quantum search.
