In quantum mechanics, symmetry operations are of importance in giving information about solutions to a system.
It is mainly used in the theoretical study of magnetic structure but is also relevant to particle physics due to CPT symmetry.
It gives basic results, the relation to ordinary representation theory and some references to applications.
Eugene Wigner[1] showed that a symmetry operation S of a Hamiltonian is represented in quantum mechanics either by a unitary operator, S = U, or an antiunitary one, S = UK where U is unitary, and K denotes complex conjugation.
Due to the presence of antiunitary operators this must be replaced by Wigner's corepresentation theory.
The equivalent of Schur's lemma for irreducible corepresentations is that the set of commuting matrices is isomorphic to
There are three cases, distinguished by the character test eq 7.3.51 of Cracknell and Bradley.
Standard representation theory for finite groups has a square character table with row and column orthogonality properties.
With a slightly different definition of conjugacy classes and use of the intertwining number, a square character table with similar orthogonality properties also exists for the corepresentations of finite magnetic groups.