We have introduced Young Tableaux, based on which we can define the irreducible symmetrizers. And we can construct the irreducible representations with these symmetrizers.

# Category: Group Theory

## Group Theory 5.2 Partitions and Young Diagrams

Gist Partition of n: A partition of the integer n is a sequence of positive integers , satisfy: .

## Group Theory 5.1 One-Dimensional Representations

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.

## Group Theory 4.3 Irreducible Operators and the Wigner-Echart Theorem

Irreducible tensors and Wigner-Eckart are important concepts in quantum mechanics, which I never master before and forget them now.

## Group Theory 4.2 The Reduction of Vectors – Projection Operators for Irrudcible Components

Based on a known irreducible representation, we can define the projection operators which can project any vectors into the irreducible space

## Group Theory 4.1 Irreducible Basis Vectors

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.

## Group Theory 3.8 Direct Product Representations, Clebsch-Gordan Coefficients

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 […]

## Group Theory 3.7 The Regular Representation

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.

## Group Theory 3.6 Orthonormality and Completeness Relations of Irreducible Characters

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.

## Group Theory 3.5 Orthonormality and Completeness Relations of Irreducible Representation Matrices

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 […]