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I'm working over an algebraically closed field of characteristic $p>0$ so I'm not assuming that $\tau$ is a direct summand of $\rho$. I think I can prove this by looking at the Kronecker product of the matrices, but does it follow from a more abstract result of representation theory?

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There is an obvious map $\tau^{\otimes n}\to\rho ^{\otimes n}$.which is injective (because it is the tensor product of several copies of the inclusion $\tau\to\rho$, and the tensor product of injections is an injection because we are tensoring over a field). What you want is an immediate consequence of this.

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    Ah, thanks for that.2012-09-24