Let $\vartheta$, $\vartheta'$ be representations of the finite group $G$ with identical characters, i.e., $$\chi(s) = \chi'(s)\ \ \ \ \forall s\in G$$.
Suppose
- $\vartheta = c_1\vartheta_1 \oplus\ldots\oplus c_n\vartheta_n$ and $\vartheta' = c_1'\vartheta'_1 \oplus\ldots\oplus c'_m\vartheta'_m$
- $\vartheta_i$ and $\vartheta'_i$ denote the irreducible inequivalent representations of $G$.
My questions are:
- $n=m$ ?
- If 1. holds, then $c_j= c'_j$, $\forall j$ ?
I am really confused about both concepts.
(My potential answer: for 1., if both representations act on the same vector space (same module), then yes. For 2., yes, since both have the same characters for all $s\in G$.)
Assuming you are talking about finite dimensional representations $\vartheta, \vartheta'$ over the complex numbers, then both questions may be answered with yes.
This follows form the orthogonality relations of characters of irreducible representations: If $\theta = \oplus_{\pi \in \widehat{G}}{c(\pi)\pi}$ is the decomposition of $\theta$ into irreducible representations $\pi$ of $G$, with $c(\pi) \in \mathbb{Z}_{\geq 0}$ denoting the multiplicity of $\pi$ in $\vartheta$, then the numbers $c(\pi)$ uniquely determined by the character $\chi_{\theta}$ through the formula $$ c(\pi) = \langle \chi_{\vartheta}, \chi_{\pi} \rangle = \frac{1}{|G|} \sum_{g \in G}{\chi_{\vartheta}(g) \overline{\chi_{\pi}(g)}}. $$ (Becuase we have $\chi_{\theta} = \sum_{\pi \in \widehat{G}}{c(\pi)\chi_{\pi}}$ and the aforementioned orthogonality realations)