Subrepresentations, Direct Sum of Representations, and Irreducible Representations
Point of post: In this post we discuss the notion of subrepresentations and irreducible representations, i.e. irreps.
This is the post which inevitably happens when discussing any new subject. It is the post where one defines the “simplest” substructures of which all other structures are “built” out of. In essence, we will discuss the concept of irreducible representations (irreps) which are the indivisible parts out of which all representations are built. To discuss this concept though we are bound by formality to build up enough machinery to state precisely what “indivisible” and “built up” means. This entails the topics of subrepresentations and the direct sum of representations which, unsurprisingly, are precisely what they sound like.
Direct Sum of Representations
Let and be finte-dimensional pre-Hilbert spaces with inner products and respectively. We can define an inner product on by
it’s trivial that this does define an inner product on . With this definition of inner product we can define the direct product of the pre-Hilbert spaces and to be the vector space with the inner product given in . We define, pursuant to earlier definitions, the direct sum of and to be the map
We note then that if and then . Indeed, for any we have that
Thus, it makes sense to define the direct sum of the representations and , denoted , to be the map
The only possible uncertainty about the above definition is whether is a homomorphism. But, this follows clearly from the simple calculation
where we’ve used the easy to verify fact that in general
1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.
2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print.