Simple idea on Irreducible Representation

We present here a good example of an irreducible representations. Studying group representations can give information about the group itself. In particular if one had an irreducible representation of a group, one can simple generate all the groups element from it.

The spin-up and -down states of the hydrogen ground state form an irreducible representation of SU(2). The element of SU(2) is no other than the Pauli matrices. It can easily be proved that the Pauli matrices satisfy the following relation

,

for

In words, this shows that each of can be rotated with all other and transformed into all other elements of SU(2).

Mathematically, matrix representation is usually states as a group representation that has no nontrivial invariant subspaces. It means, for the element of the group, say V the subgroup can either be zero (i.e. trivial) or the group itself ( and W=V). Note that, most of the textbook usually used the term nontrivial for and .

Remarks: We have discuss a bit on the group representation theory here.
Reference: 1. Weber and Arfken, Essential Mathematical Methods for Physicists 2. B.C. Hall, Lie Groups, Lie Algebras & Representations

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