Inner product of functions
This definition works in all non-modular characteristics and is bilinear.
Let be a finite group and be a field whose characteristic does not divide the order of . Given two functions , define:
Note that makes sense as an element of because is finite and the characteristic of does not divide the order of .
Hermitian inner product
This definition works over or any subfield of that is closed under complex conjugation.
Let be such a field and be a finite group. Given two functions , define:
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 of a representation of a finite group and an element , . 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.