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

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

*Irreducible Representations*

Let be a finite dimensional pre-Hilbert space and a finite group. A representation is called *irreducible *if the only subspaces of invariant are and . 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 is irrep if and only if is not equivalent to the direct sum of two non-trivial representations. (a *non-trivial *representation is a representation such that ).

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

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

**Proof: **We proceed by induction on . If 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 such that and let be some representation with . If is an irrep we’re done, and if not then for some representations and with thus by assumption we have that and and thus evidently (it’s fairly obvious, and easy to prove) we may conclude that

from where the conclusion follows.

Clearly the relation is an equivalence relation on the set of irreps on a group . We denote the set of all such equivalence classes as . 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.

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