Inner product of functions

Definition

Bilinear form

This definition works in all non-modular characteristics and is bilinear.

Let $G$ be a finite group and $k$ be a field whose characteristic does not divide the order of $G$. Given two functions $f_{1},f_{2}:G\to k$, define:

$\langle f_{1},f_{2}\rangle _{G}={\frac {1}{|G|}}\sum _{g\in G}f_{1}(g)f_{2}(g^{-1})$

Note that $1/|G|$ makes sense as an element of $k$ because $G$ is finite and the characteristic of $k$ does not divide the order of $G$.

Hermitian inner product

This definition works over $\mathbb {C}$ or any subfield of $\mathbb {C}$ that is closed under complex conjugation.

Let $k$ be such a field and $G$ be a finite group. Given two functions $f_{1},f_{2}:G\to k$, define:

$\langle f_{1},f_{2}\rangle _{G}={\frac {1}{|G|}}\sum _{g\in G}f_{1}(g){\overline {f_{2}(g)}}$

Note that this is a Hermitian positive-definite inner product.

Relation between the definitions

First, note that the two inner products defined are not the same thing over a field where both definitions are applicable. The former is bilinear while the latter is sesquilinear and positive-definite (these qualities make it Hermitian). However, the following is true:

For a character $\chi$ of a representation of a finite group $G$ and an element $g\in G$, $\chi (g^{-1})={\overline {\chi (g)}}$. See trace of inverse is complex conjugate of trace.

Thus, if we apply the inner product operation only to characters of representations, then both definitions work exactly the same way.