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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 $\mathbf C$.

This is a copy of this question of mine on MSE, which appears to have gone largely unnoticed.

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    $\begingroup$ What do you actually want to know? If $G$ is abelian then an "inner product" on $G$ could just be thought of as an isomorphism from $G$ to its dual. But I don't really see a maths question here. You can make any definition you like but the proof of the pudding is in the eating (is that a UK expression??) $\endgroup$
    – Kevin Buzzard
    Commented Jan 31, 2017 at 21:15
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    $\begingroup$ The choice of the target of the inner product is already not clear. Of course you can map into the base ring, but there are other natural choices (e.g, for a local ring, the injective hull of its residual field; in the case of $\mathbb{Z}$, $\mathbb{Q}/\mathbb{Z}$, etc). $\endgroup$
    – YCor
    Commented Jan 31, 2017 at 21:25
  • $\begingroup$ @Kevin Thus (reference-request): I’m searching for a unified treatment of inner products on the different kinds of stuff one can apparently define them on, the choice of stuff being motivated by the “eating” of a meaningful set of theorems. (Re: choosing $\hat G \cong G$, yes, a general description of duality would be another way to put it—but I don’t know how to make it include the representation ring example. It is funny, though, how the $\hat G \cong G$ description automatically yields that the target of the product for finite groups is $\mathbf Z/e\mathbf Z$.) $\endgroup$
    – Alex Shpilkin
    Commented Jan 31, 2017 at 21:40
  • $\begingroup$ @YCor Well, it could be that there is no unique general theory. I’m only trying to figure out if anyone actually cared enough to pose the question. $\endgroup$
    – Alex Shpilkin
    Commented Jan 31, 2017 at 21:50

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