I am currently studying the book Lie Algebras in Particle Physics by Howard Georgi (2nd edition) and I couldn't understand this part:
A representation is completely reducible if it is equivalent to a representation whose matrix elements have the following form:
$$\begin{bmatrix}D_1(g) & 0 & 0\ … \\0&D_2(g) &0\ … \\ …\ & …\ & … \end{bmatrix}$$
where $D_j(g)$ is irreducible for all $j$. This is called block diagonal form.
Notation: $g$ refers to an element of a group $G$ and $D$ is a representation.
What I don't quite get is: Those diagonal elements are just numbers aren't they? Then what's with calling them irreducible representations? Going further, the author says:
A representation in block diagonal form is said to be the direct sum
of the subrepresentations, $D_j$s
I am a beginner so excuse me if this is trivial. Why exactly did he say that those matrix elements are representations themselves? What am I missing?
Best Answer
The elements on the diagonal $D_j(g)$, are not just numbers. They are in general matrices. That is why it is called block diagonal. As a very simple example, take SU(2): the tensor product of two SU(2) matrices, each being in the two-dimensional representation, would give you a four-dimensional matrix. Using the appropriate transformation of this matrix, one can reduce it to a block diagonal form with a three-dimensional matrix and a one-dimensional element sitting on the diagonal. The three-dimensional matrix represents the adjoint representation of SU(2) and the one-dimensional element is the singlet representation. Does this help?