We have introduced Young Tableaux, based on which we can define the irreducible symmetrizers. And we can construct the irreducible representations with these symmetrizers.
Symmetry group is just the permutation group, which has a non-trivial invariant subgroup consisting of all even permutations. (An even permutation is one which is equivalent to an even number of simple transpositions.) has a coset consisting of all odd permutations.
Whenever a physical system possesses a symmetry, the basic solutions to the classical equations or the basis state vectors of the corresponding quantum mechanics equations problem are naturally classified according to the irreducible representations of the symmetry.
Direct Product Vector Space: Let and be inner product spaces and and be orthonormal basis in the two spaces respectively. Then the direct product space consists of all linear combinations of the orthonormal basis vector , which can be regarded as , such that (i) Inner product: ; (ii) where the complex number are the […]
OK, I still don’t known the physics applications with all of the irreducible representation. In the last section, we can deduce how many irreducible representation are consisted in a group representation, but what’s the meaning? The real answer may be clear if we have a reality application of group representation.
In the previous section, we have developed the orthonormality and completeness relations of irreducible representation matrices. The matrices is complex with so many elements, we will study the relations fo characters, which will simplify our study of the irreducible representations.
As we have said, for the mathematical simplicity, we will focus on irreducible representation. We also found that most of the representation is equivalent to a unitary representation. Based on Schur’s Lemmas, we will encounter the central results of group representation theory, which related the inequivalent irreducible representations together, although I still don’t know what’s […]