  Tower of States analysis is a powerful tool for investigating phase
transitions in condensed matter systems. Spontaneous symmetry breaking implies
a specific structure of the energy eigenvalues and their corresponding quantum
numbers on finite systems. In these lecture notes we explain the group
representation theory used to derive the spectral structure for several
scenarios of symmetry breaking. We give numerous examples to compute quantum
numbers of the degenerate groundstates, including translational symmetry
breaking or spin rotational symmetry breaking in Heisenberg antiferromagnets.
These results are then compared to actual numerical data from Exact
Diagonalization.
