Characters and decomposition

Question

This is taken from Linear representations of finite groups by J.P.Serre, page 16. My query is :

1.Does distinct irreducible characters imply non-isomorphic?

2.Here in the decomposition of $V$, $W_{i}$ is non-isomorphic to $W_{j}$ for $i\ne j$ right?

Answer 1

A character $\chi: G \to \mathbb{C}$ of a $G$-representations is always constant on conjugacy classes, that means $$\forall g,h \in G: \chi(hgh^{-1})=\chi(g)$$

So all the characters lie in the space $T = \{ \chi: G \to \mathbb{C}: \chi \text{ constant on conjugacy classes}\}$. The number of conjugacy classes of $G$ is therefore the dimension of $T$.

One can show that the irreducible characters are linear independent and that the number of simple $G$-modules is the number of conjugacy classes of $G$, so the irreducible characters are a basis of $T$. We get the following:

Let $V, W$ be $G$-modules with representations $D_v: G \to GL(V), D_w: G \to GL(W)$ and characters $\chi_V, \chi_W: G \to \mathbb{C}$, then

$$V \simeq W \Leftrightarrow \chi_V = \chi_W$$

While the direction $\Rightarrow$ is simple (see the comments), the $\Leftarrow$ is also not difficult now:

Using the notation above, let $\chi_V = \chi_W$. According to Maschke and Schur there is a unique decomposition of $V$ and $W$ into the simple $G$-modules $W_1, ..., W_r$:

$V = m_1 W_1 \oplus ... \oplus m_r W_r$

$W = m'_1 W_1 \oplus ... \oplus m_r' W_r$

where the $m_i$'s and the $m'_i$'s are the multiplicities, means $m_i W_i = W_i \oplus ... \oplus W_i$ ($m_i$ times).

$\chi_v = \chi_w \Rightarrow \sum_{i=1}^r m_i \chi_{W_i} = \sum_{i=1}^r m'_i \chi_{W_i} \Rightarrow m_i = m'_i f.a. i=1,...,r \Rightarrow V \simeq W$,

where the secound implication follows from the linear independence of the irreducible characters.