Different particles can be represented as different irreducible representations of Poincare group. Can we classify quasiparticles using irreducible representations of some group? Also, quasiparticles are low energy excitations of the system, and it is closely related to the ground states of the system, so I am wondering if there is any relations we can use to help us classify quasiparticles from ground state symmetry. Also, is the irreducible representation of some group argument can apply to those excitations of topological origin?
The group representation argument can not be applied to those excitations of topological origin. Group theory is not the most general classification criterion of quasiparticles. Group theory is useful when symmetry plays an important role in the discussion. For example, Goldstone modes in the symmetry breaking phases can be classified by group representation theory, boundary modes in the symmetry protected topological phases can be classified by group cohomology theory. But if the system has no symmetry (or the symmetry is not relevant to the discussion), then group theory may not be very useful. For example, different anyon excitations in topologically ordered phases are not classified by their group representations: they may all belong to the trivial representation for example, but still, they are distinct topological excitations and can not transit from one to another. In this case, we use category theory to classify different topological excitations. In fact, category theory is a more general classification criterion of quasiparticles. Because the representations of a group also form a category, so the group representation classification is part of the categorical classification. Moreover, the categorical classification also captures the topological excitations nicely, which demonstrate its advantage over the group theory classification.