I have several questions about symmetry in quantum mechanics. It is often said that the degeneracy is the dimension of irreducible representation. I can understand that if the Hamiltonian has a symmetric group G, then the state space with the same energy eigenvalue will carry a representation of G. However, why this representation is usually irreducible? Wigner proved that any symmetric group representation in quantum mechanics must be either unitary or anti-unitary. Is it true that the representation of continuous symmetric group must be unitary and cannot be anti-unitary? What is the difference between geometric symmetry and dynamical symmetry? By dynamical symmetry I mean for example the SO(4) symmetry of hydrogen. Some text refers dynamical symmetry to "internal" symmetry. How to state the definition of dynamical symmetry strictly? If it can't be obtained by rotation, translation and inversion, then how do to find it generally?