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In the paper I called it the Burnside decomposition, but it looks like the standard name is the Wedderburn decomposition. That might simply have been a mistake in terminology on my part. Anyway, there are two good ways to get the summands to be $V \otimes V^*$. (Of course they are closely related.) 1) You can interpret $\mathbb{C}[G]$ as an associative ...

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