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Consider the permutation group $S_3$. Maschke's Theorem says that the group algebra $\mathbb{C}S_3$ is semisimple, i.e., isomorphic to a direct sum of matrix algebras. I want to find matrix units for $\mathbb{C}S_3$ (those elements corresponding to the matrices with a single entry of 1 and zeros everywhere else) but I'm not sure how to do this.

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  • $\begingroup$ I actually tried to do this recently but gave up before finding them all, since it gets more complicated than I was prepared for (unless there are some more tricks I am missing). You know you are looking for 3 elements (since this is the number of conjugacy classes) and these will be central idempotents that sum to $1$ in the group algebra. Now, a good place to start is that any normal subgroup gives you a central idempotent. $\endgroup$ – Tobias Kildetoft Jul 26 '17 at 8:50
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We have $\mathbb{C}S_3 \cong \mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C}^{2 \times 2}$ by Artin-Wedderburn. In fact, an isomorphism is given by $$\tau: g \mapsto (1,\text{sgn}(g), \rho(g))$$ where $\rho : \mathbb{C}S_3 \to \mathbb{C}^{2 \times 2}$ is a 2-dimensional irreducible representation (permutation representation modulo fixed points).

Let us first find the central idempotents in $\mathbb{C}S_3$, corresponding to $(1,0,0)$, $(0,1,0)$, $(0,0,I_2)$.

In general, if $\chi$ is an irreducible character of a finite group $G$, the element $$ e_\chi = \frac{\chi(1)}{|G|} \sum_{g \in G} {\chi(g^{-1})} g $$ of $\mathbb{C}G$ is the corresponding central idempotent.

So $e_1 = \frac{1}{6}\sum_{g \in S_3} g$ corresponds to $(1,0,0)$ and $e_\text{sgn} = \frac{1}{6}\sum_{g \in S_3} \text{sgn}(g) g$ corresponds to $(0,1,0)$. Also, $e_\rho = 1 - (e_1 + e_\text{sgn})$ corresponds to $(0,0,I_2)$.

Now we really have to look at the representation $\rho$ and find a preimage of each of the four matrix units in $\mathbb{C}^{2 \times 2}$. These preimages may not be the matrix units of $\mathbb{C} S_3$ themselves, but you can just multiply them with $e_\rho$ to obtain the desired matrix units (since $\rho|_{\mathbb{C}S_3 e_\rho} : \mathbb{C}S_3 e_\rho \to \mathbb{C}^{2 \times 2}$ is an isomorphism).

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