# Abstract Nonsense

## Subrepresentations, Direct Sum of Representations, and Irreducible Representations (Pt. III)

Point of post: This post is a continuation of this one.

Irreducible Representations

Let $\mathscr{V}$ be a finite dimensional pre-Hilbert space and $G$ a finite group. A representation $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ is called irreducible if the only subspaces of $\mathscr{V}$ invariant are $\mathscr{V}$ and $\{\bold{0}\}$. It is often the case, and we will follow this here, that an irreducible representation is shortened to irrep. It’s clear from the last two theorems in our last section that $\rho$ is irrep if and only if $\rho$ is not equivalent to the direct sum of two non-trivial representations. (a non-trivial representation $\omega$ is a representation such that $\deg\omega>1$).

We now make clear why irreps are the “building” blocks of representations, we do this via the next theorem:

Theorem: Let $G$ be a finite group. Then, any non-trivial representation on $\rho$ is equivalent to the direct sum of irreducible representations.

Proof: We proceed by induction on $\deg\rho$. If $\deg\rho=1$ this is trivial since every representation of degree one is an irrep (it can’t have a non-trivial invariant subspace since it can’t have any non-trivial subspace!). Assume that it’s true for all $\rho$ such that $\deg\rho and let $\omega$ be some representation with $\deg\omega=n$. If $\omega$ is an irrep we’re done, and if not then $\omega\simeq \rho\oplus\rho'$ for some representations $\rho$ and $\rho'$ with $0<\deg\rho,\deg\rho' thus by assumption we have that $\displaystyle \rho\simeq \nu_1\oplus\cdots\oplus \nu_k$ and $\rho'\simeq \upsilon_1\oplus\cdots\oplus\upsilon_\ell$ and thus evidently (it’s fairly obvious, and easy to prove) we may conclude that

$\omega\simeq \nu_1\oplus\cdots\oplus\nu_k\oplus\upsilon_1\oplus\cdots\oplus\upsilon_\ell$

from where the conclusion follows. $\blacksquare$

Clearly the relation $\simeq$ is an equivalence relation on the set of irreps on a group $G$. We denote the set of all such equivalence classes as $\widehat{G}$. This will be of the utmost importance in the theory to come.

References:

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.