# Are “FG-module characters” sometimes used, too?

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

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.