Let $G$ be a finite group and $\mathbb{C}G$ its associated group algebra. The regular representation $\mathbb{C}G\subset B(\ell^2(G))$ defined by:
$$\delta^g(e_{g_2})=e_{g_1g_2},$$
contains each irreducible representation $\mathbb{C}G\subset B(V_\alpha)$ with multiplicity equal to its degree.
In the case of $\mathbb{C}S_3$, where $V_0$ is associated to trivial representation, $V_1$ to the sign representation, and $V_2$ to the degree two irreduciblee representation, the regular representation is a faithful representation of $\mathbb{C}S_3$ on $\mathbb{C}^6\cong V_0\oplus V_1\oplus V_2\oplus V_2$.
However there is also a faithful representation of $\mathbb{C}S_3$ that just uses each irreducible representation once.
Question: Do we always have a faithful representation when we just use every irreducible representation just once?
Best Answer
HINT:
If $\rho$ contains all the irreducible representations, then $k(G)\subset \rho^{\oplus n}$ for some $n$. If $\rho$ were not faithful, then so would be $\rho^{\oplus n}$, contradiction.