  The problem of construction of projection operators on eigen-subspaces of
symmetry operators is considered. This problem arises in many approximate
methods for solving time-independent and time-dependent quantum problems, and
its solution ensures proper physical symmetries in development of approximate
methods. The projector form is sought as a function of symmetry operators and
their eigenvalues characterizing the eigen-subspace of interest. This form is
obtained in two steps: 1) identification of algebraic structures within a set
of symmetry operators (e.g. groups and Lie algebras), and 2) construction of
the projection operators for individual symmetry operators. The first step is
crucial for efficient projection operator construction because it allows for
using information on irreducible representations of the present algebraic
structure. The discussed approaches have promise to stimulate further
developments of variational approaches for electronic structure of strongly
correlated systems and in quantum computing.
