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I am only beginning my study of group representations and characters. So far I have already encountered the regular group algebra $FG$. Although in an FG-module the multiplication is only defined for elements of the group $G$, it has unique linear extension, so in fact the regular module $FG$ acts on any $FG$-module, see e.g. Gordon-Liebeck, Representations and Characters of Groups p.57. And there are several situations where it is useful that we can use elements from $FG$.

At the moment I am starting to read about characters in the book I've mentioned above. Characters are functions from $G$ to $\mathbb C$. Of course, any such function has unique linear extension to a linear function $\mathbb C G\to\mathbb C$ defined on the regular $\mathbb C G$-module. So I was wondering, whether there are situations, where this extension comes handy.

Are group characters sometimes studied as linear maps on the regular $\mathbb C G$-module $\mathbb C G$? Are there some situations, where this approach can simplify things?

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I have not seen this done, and my guess as to the reason is this: The reason we can make good use of the group algebra is that it has a multiplication that is compatible with our representation. For a character, this is no longer the case. –  Tobias Kildetoft Feb 7 '13 at 14:56

2 Answers 2

I doubt this approach simplifies anything for the following reason: As a vector space $FG$ is free on the set $G$ so there is a canonical and natural bijection between linear maps $FG \to F$ and set maps $G \to F$. So there is a technical sense in which there is no additional information present by considering the linear map induced by a character.

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Certainly Jim has some reasons. But I think there is still some reason behind this algebra consideration:
First of all, even though the sets of characters of a group algebra and of a group are in bijective correspondence, the two sets might still contain a different amount of information. For example, the conjugacy class sums form a basis of the center of this group algebra, and in turn they correspond bijectively to irreducible characters. So, when interpreted over the algebra, one could view (irreducible) characters as corresponded to the center of the algebra.
Secondly, after one extends the structure of a (finite) group to a (semisimple) algebra, one of course gets more things from the structure of this algebra, like its multiplicative constants and so on.
In fact, as to the fact that the degrees of a character must divide the order of the group, I think the best way to look at it is by use of this algebraic tool.
Hope this helps.

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