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This is equivalent to the statement that any irreducible representation of a group $G$ is contained in some tensor power of a faithful representation $V$.

A proof, due to Brauer, is given e.g. here. In fact, it is possible to make the size of the tensor power that one has to take explicitly bounded (the number of distinct values taken by the character of $V$ minus 1), using vanderMonde determinants.

I learned the theorem from Curtis and Reiner, Methods of Representation Theory.

In the case of $GL(V)$, it doesn't work because $V$ is a faithful representation, but not every irreducible of $GL(V)$ is contained in a tensor power (you have to look at the duals as well).

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This is equivalent to the statement that any irreducible representation of a group $G$ is contained in some tensor power of a faithful representation $V$.

A proof, due to Brauer, is given e.g. here. In fact, it is possible to make the size of the tensor power that one has to take explicitly bounded (the number of distinct values taken by the character of $V$ minus 1), using vanderMonde determinants.

In the case of $GL(V)$, it doesn't work because $V$ is a faithful representation, but not every irreducible of $GL(V)$ is contained in a tensor power (you have to look at the duals as well).