Inner product of functions

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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.