# Repeated Irreducible Representations in a representation

I'm reading through Serre's - Linear representations of finite groups. He has the following theorem (theorem 4 of chapter 2):

Let $V$ be a linear representation of $G$, with character $\phi$ and suppose that $V$ decomposes into a direct sum of irreducible representations:

$V=W_{1}\oplus\cdots\oplus{W_{k}}$

Then, if $W$ is an irreducible representation with character $\chi$, the number of $W_{i}$ isomorphic to $W$ is equal to the scalar product $(\phi|\chi)=\langle\phi,\chi\rangle$.

I'm a little confused by this. The conditions for $V=W_{1}\oplus{W_{2}}$ is every $x\in{V}$ can be written $x=w_{1}+w_{2}$ with $w_i\in{W_i}$ and $W_1\cap{W_2}=\{0\}$. So wouldn't the fact that you can decompose $V$ in to a direct sum as above mean that any $W_{i}$ appears at most once? Is there some form of distinction between an internal and external direct sum involved that I'm missing or have I just misunderstood the concepts involved?

• Read: "$W_i$ isomorphic to $W$" Commented Jul 21, 2014 at 12:16
• @MartinBrandenburg: I'm sorry but your comment doesn't really clarify things. Commented Jul 21, 2014 at 12:23
• If you forget about representation theory for a minute - an $n$-dimensional vector space over $K$ decomposes as $V=V_1\oplus\dotsb\oplus V_n$, where each $V_i$ is isomorphic to $K$. So all of the summands are isomorphic to each other as vector spaces, despite being different subspaces of $V$. Given this, it shouldn't be so hard to believe that you can have a representation $W=W_1\oplus W_2$ with $W_1$ and $W_2$ isomorphic representations, even though $W_1$ and $W_2$ are as disjoint as possible inside $W$.
– mdp
Commented Jul 21, 2014 at 12:37

It might help to consider the case when the group $G$ is the trivial group, i.e. $G = \{1\}$. Then giving a $G$-representation is just the same as giving a vector space (the $G$-action is always via the identity).

• An irreducible representation is just a one-dimensional vector space.

• Any vector space is a direct sum of one-dimensional vector spaces.

• All one-dimensional vector spaces are isomorphic.

• Not all vector spaces are one-dimensional.

Does this clarify anything?

• OK. I think I kind of sort of see the point you are trying to make. Leaving representations for a bit, this is like looking at $<(1,1,0)>$ and $<0,1,1>$ inside of $\mathbb{R}^3$. They are both isomorphic to $<(1,1)>$ in $\mathbb{R}^2$ but they are not the same thing inside of $\mathbb{R}^3$ (all of these taken as vector spaces over $\mathbb{R}$). But then according to what you say wouldn't all the $W_{i}$ be isomorphic to each other? (by your first three bullet points?). Where does the difference come in? Commented Jul 21, 2014 at 12:42
• @DanulG: For the trivial group, yes, all the $W_i$ would be isomorphic. But if $G$ is not trivial, it admits more than one irreducible rep'n (but only finitely many non-isomorphic ones, if its finite), and so the $W_i$ can be any representations for this list of isomorphism classes of irreps. Commented Jul 21, 2014 at 16:23