Linear combination of a character

Question

Hi there I am trying to solve a problem about characters of a finite group $G$. If $\chi$ is a character such that $\langle \chi,\chi \rangle = 2$ and $\chi_1,\dotsc,\chi_n$ the irreducible characters of $G$ we need to find $\chi$ as an $\mathbb{N}$-linear combination of $\chi_i$'s. I thought of taking the inner product $\langle \chi,\chi \rangle$ but I can't continue further. Then maybe with orthogonality relations but I don't have enough for this. Any ideas? I am sorry if this is a bad question but I just started studying non-commutative algebra. Thanks for your help.

Answer 1

The only you can deduce is that $\chi=\chi_i+\chi_j$ for some $1\leq i\not=j\leq n$.
If we write $\chi=\sum_{i=1}^nn_i\chi_i$ for some nonnegative integers $n_i$ then, since $\langle \chi,\chi \rangle=2$ it follows that $$2=\langle \chi,\chi \rangle=\langle\sum_{i=1}^nn_i\chi_i,\sum_{j=1}^nn_j\chi_j\rangle=\sum_{i,j=1}^nn_in_j\langle\chi_i,\chi_j\rangle=\sum_{i=1}^nn_i^2.$$ This implies that there are only two nonzero coefficients, say $n_i,n_j$ (with $i\not=j$), with value $1$ and that all the other coefficients are necessarily $0$. Hence the claim follows.