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How do I find the projection of a representation onto a direct sum of irreducible representations?

For example, how do I find the projection of the three-dimensional "defining representation" of the symmetric group of order 3 ($S_3$) onto the representation $\textbf{1} \oplus \textbf{2}$, where $\textbf{1}$ and $\textbf{2}$ are the trivial representation and the two-dimensional irreducible representation of $S_3$, respectively.

I know there is the formula $P_a=\frac{n_a}{n_G}\sum_g \chi_{D^\dagger_a}(g)^* D(g)$ but it is to project a generic representation onto an irreducible representation. Can I apply it also to a direct sum representation?

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  • $\begingroup$ But the standard rep of $S_3$ is an internal direct sum of $\bf1$ and $\bf2$ subreps. So it sounds like you're asking for a projection from a vector space to itself, which the identity function is an example of, no? What am I missing? Did you mean to ask for projecting onto the $\bf1$ and $\bf2$ subreps? Also note that every complex rep of a finite group is an internal direct sum of irreps, and the formula you cite for projecting onto isotypical components works for any rep - so asking if you can apply the formula for direct sums is extremely weird to ask, no? $\endgroup$ – runway44 Jan 14 at 10:38
  • $\begingroup$ Does this answer your question? How to Decompose Representations into Irreducible Ones? $\endgroup$ – ahulpke Jan 14 at 14:12

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