# question on group representations

Here is a problem I faced in algebra. $\rho: A_4 \rightarrow End_{\mathbb C}\mathbb C^{10}$ is a representation of $A_4$. Then show that there is a vector $v \in \mathbb C^{10}$ such that $v$ is an eigenvector for all $\rho(g)$, $g \in A_4$. I think I miss some trick, or something. Any idea will be appreciated.

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The question is not clear. Are you trying to prove that if $\rho$ is a representation... then there is a vector $v$ ...? – Gerry Myerson Aug 17 '13 at 6:37
@GerryMyerson you are right, I edited the question! – user51128 Aug 17 '13 at 6:51

A vector $v \in \mathbb C^{10}$ is an eigenvector for all $\rho(g)$ if and only if $\mathbb Cv$ is a one dimensional invariant subspace. So you want to show that every representation of dimension $10$ must have a $1$ dimensional irreducible summand. If you compute the dimensions of the irreducible representations of $A_4$ you'll find that they are $1$, $1$, $1$, and $3$. As $3$ does not divide $10$ one of the summands of your representation must have dimension $1$.
+1 @user51128: I would be very surprised, if the dimensions of the irreducible representations of $A_4$ (and $S_4$) had not been calculated in an example/exercise prior to asking this question in any textbook on representation theory. – Jyrki Lahtonen Aug 17 '13 at 7:52