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.
What is Linear Transformation? Mapping: If are two non-empty sets, and there exists a rule which make every element in corresponds one unique element in . Then the rule is called a mapping. Transformation: If is a linear space. The mapping from to itself is called a transformation. Linear Space: If we have a […]
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 […]
We know that some representation may be reducible, but how can we find out whether it is full reducible. If we can reduce the representation into some lower dimension, it’ll be very helpful for our to discuss specific groups. Fortunately, we find that unitary group will be full reducible or irreducible. And the unitary group […]
Gist Representation is just a homomorphism mapping (1) where is an operator on linear vector space , such that (2) Dimension of the representation: is the dimension of the vector space . Faithful: If the homomorphism is also an isomorphism. Degenerate representation: The one which is not Faithful. Matrix Representation: is the case […]
We can use group representation to discuss group as it’s said in section 3.1 now. But there may be so many representation, which one will be the simple one without erasing the details of the group. This section is to discuss the simplicity.
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