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In Fulton and Harris's Representation Theory: A First Course, the authors describe an isomorphism $\phi: \mathbb{C}G \to \bigoplus \text{End}(W_i)$ as follows:

As we have said, for any representation $W$ of $G$, the map $G \to \text{Aut}(W)$ extends by linearity to a map $\mathbb{C}G \to \text{End}(W)$; applying this to each of the irreducible representations $W_i$ gives us a canonical map $$\phi: \mathbb{C}G \to \bigoplus \text{End}(W_i)$$ This is injective since the representation on the regular representation is faithful. Since both have the dimension $\sum (\dim(W_i))^2$, the map is an isomorphism.

I do not quite understand a comment made shortly after that:

Next, we can think of $\phi$ as the decomposition of the semi simple algebra $\mathbb{C}G$ into a product of matrix algebras. It implies that the matrix entries of the irreducible representations give a basis for the space of all functions of $G$

My questions are these:

  • This seems to give $\mathbb{C}G$ as the direct sum of some algebras. How does this imply that it is the product of matrix algebras.
  • How does this imply that the matrix elements of the irreducible representations give a basis for the space of all functions on $G$.
  • I know that this is also an orthogonal basis. Can we get result from this as well?
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    $\begingroup$ In this case, the direct sum and product coincide since they're both finite. You can verify that they both fulfill the same universal properties $\endgroup$ – leibnewtz Nov 4 '17 at 22:44
  • $\begingroup$ If $\dim W=d$ then ${\rm End}(W)\cong M_d(\Bbb C)$ is a matrix algebra (up to isomorphism). The matrix coefficients (as functions on $G$, interpreted as elements of $\Bbb C[G]$) correspond to the obvious matrices in these matrix algebras under $\phi$. $\endgroup$ – anon Nov 6 '17 at 13:52

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