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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 ...
To see that Simon's program is an instance of an (abelian) hidden subgroup problem, we have to identify the group $G$, the subgroup $H$, the set $X$ and the function $f : G \rightarrow X$. Note first that the set $\{ 0,1 \}^n$ of all bit vectors of length $n$ naturally comes with a group structure given by the (component-wise) XOR between bit vectors: $(x_1, ... 3 Two classical texts for the representation theory of finite groups are the books of Hamermesh and Serre. These books however lack chapters on Fourier analysis needed for the quantum computation applications. For a more modern text for finite group representations which includes a chapter on Fourier analysis, please see the lecture notes by: Steinberg. ... 3 The second part of each tensor product serves as a multiplicity space. It might be more satisfying to write it as a full decomposition like your second one. So you have$\bigoplus V_\lambda \otimes V_\lambda^*$and you want it to look more like your second equation. What would happen is each$V_\lambda$would show up as a direct summand$d_\lambda$times ... 3 The question is whether taking the Fourier transform$\operatorname{QFT}|gH\rangle$followed by sampling allows to efficiently recover generators of the hidden subgroup$H\leq G$. While the problem is wide open for non-abelian groups (see this paper for a discussion of the limitations of the Fourier sampling method for instances in case of$G=S_n$,$G=PSL(2,\...