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

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

*Invariant Subspaces and Subrepresentations*

Let be a finite-dimensional pre-Hilbert space and a finite group. Let be a representation. Then, we call *invariant under *if is invariant under in the normal sense for each . It’s clear then that the map

is a representation of on . Such a representation is called a *subrepresentation *and is denoted or *. *Our first result concerning invariant subspaces and subrepresentations is a fundamental one. Namely:

**Theorem: ***Let be a finite dimensional pre-Hilbert space with inner product and a representation of on . Suppose that is invariant under then so is *

*Moreover, .*

**Proof: **Fix , and let and be arbitrary. We note then that

and since were arbitrary it follows that for every . But, by the arbitrariness of we may conclude that for every and . Said differently is invariant under .

Now, to prove that we define

where is the unique representation of every element of as the sum of an element of and an element of (this comes from the elementary fact that where now the denotes the internal direct sum). This is evidently a unitary map since (recalling that by definition)

We then note that for every and we have that

from where it follows that .

We also have somewhat of a converse of this theorem:

**Theorem: ***Let be a finite dimensional pre-Hilbert space and a finite group. Then, if is a representation of on such that where*

* *

*and*

*then if is the guaranteed unitary isomorphism such that for every then is invariant under .*

**Proof: **Let then we have that . Since and were arbitrary the conclusion follows.

**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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