Properties of generalized characters

Question

I search for generalized characters which are not characters.Also I want to know that why every generalized character is a difference of characters.

Answer 1

Assuming you mean a generalized character $\chi$ is a linear combination $\chi = \sum_{i = 1}^n d_i \chi_i$ of characters $\chi_i$ with coefficients $d_i \in \mathbb{Z}$, it follows that a generalized character is a character if and only if $d_i \geq 0$ for all $i$. Assume that $\chi$ is not a character, so that $d_i < 0$ for some $i$. Reindexing the $\chi_i$, we may suppose that there is an $m \in \{1,\ldots,n\}$ such that $d_i \geq 0$ for $i \leq m$ and $d_i \leq 0$ for $i > m$. Set $$ \alpha = \sum_{i=1}^m d_i\chi_i $$ and $$ \beta = -\sum_{i=m+1}^n d_i\chi_i. $$ Then $\alpha$ and $\beta$ are generalized characters with all positive coefficients, and so $\alpha$ and $\beta$ are actually characters. Also, it follows immediately from their definitions that $$ \chi = \alpha - \beta. $$