# Inner products on abelian groups and general modules

Where can I find a discussion about inner products on something more general than vector spaces, and to which extent should one attempt such a generalization?

My particular interest is in abelian groups (specifically, the representation ring of a finite group, as described on MO). The only references in this case seem to be Distance-Regular Graphs by Brower, Cohen and Neumaier (1989), p. 72, which does provide a construction for finite groups (contingent on their classification, or so it seems), but no connection to the field case, and λ-Rings and the Representation Theory of the Symmetric Group by Knutson (1973), which only concerns itself with the representation ring. (See also this post on Ask an Algebraist about the former book.) I would like to see how it fits with the usual inner products over $\mathbf R$ and $\mathbf C$.

Seeing as this question appears to have gone largely unnoticed, I have posted a copy on MO.

• This there any context to this? What are you trying to understand? What motivation do you have for generalizing?
– JMag
Jan 31 '17 at 22:11
• @JMag The representation ring “inner product” $(V,W) = \mathop{\mathrm{dim}} \mathop{\mathrm{Hom}}(V,W)$ that turns into the inner product of characters when tensored with $\mathbf C$. The MO link and Knutson reference in the question describe this idea. Feb 1 '17 at 0:19