# Multiplicity of a completely reducible representation in another irreducible representation.

I have got the next question that I am pondering the answer to.

Let $\tau$ be a completely reducible representation of finite dimension of a group $G$, and let $\pi$ be another irreducible representation of finite dimension of $G$. Suppose these representations are over a closed algebraic field, $K$.

I want to show that $\pi$'s multiplicity in $\tau$ equals $dim_K(Hom_G(\pi,\tau))$, where $T\in Hom_G(\pi,\tau) \Leftrightarrow \forall g \in G\ T\pi(g) = \tau(g) T$.

Do you have any insight or good reference for this question?

Basically, if $\tau \cong n_1 \pi_1 \oplus \cdots \oplus n_k \pi_k$, and $\pi \cong m_i\pi_i$ for some $1\leq i\leq k$, then I want to show that $$m_i=dim_K(Hom_G(\pi,\tau))$$

I am not sure how to show this? any good refernece or better yet hint to the right way?

Thanks in advance.

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Hint: Schur's lemma. – user27126 Dec 4 '12 at 18:29
The title of your question is the reverse of your actual question, i.e., you're asking about the multiplicity of the irreducible in the reducible representation. – Keenan Kidwell Dec 4 '12 at 18:45

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