Projection of a group representation onto a direct sum of irreducible representations

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?

• 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? – runway44 Jan 14 at 10:38
• Does this answer your question? How to Decompose Representations into Irreducible Ones? – ahulpke Jan 14 at 14:12