Question about inner product of functions from a group to a field.

Question

In The Symmetric Group the author, Bruce Sagan, writes on page 35 that the definition of the inner product is $$ \langle\chi,\psi\rangle=\frac{1}{|G|}\sum\limits_{g\in G}\chi(g)\psi(g^{-1}), $$ where $\chi, \psi$ are functions from a group $G$ to an arbitrary field. However, he goes on to write 'but over the complex numbers this "inner product" is only a bilinear form'. I do not understand what he means by that. It seems like he starts by saying that we define the inner product for an arbitrary field, but it does not apply to the complex numbers, which is quite confusing. Can someone help me understand what he means?

Answer 1

You are rightly confused here. It happens that Sagan uses the expression "inner product" in a non standard way. Usually, inner products are only defined over $\mathbb{R}$ or $\mathbb{C}$. Over $\mathbb{R}$ an inner product is a bilinear map $B$ from $V\times V$ (where $V$ is a real vector space) into $\mathbb{R}$ which is symmetric and such that $B(v,v)>0$ when $v\in V\setminus\{0\}$. But the definition is different over $\mathbb{C}$. Then $B$ must be sesquilinear (meaning that it preserves sums for each variable and that $B(\alpha v,\beta w)=\alpha\overline\beta B(v,w)$) and conjugate symmetric (meaning that $B(w,v)=\overline{B(v,w)}$).