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In several texts it is standar to define the group algebra $\mathbb{C}[G]$ of a group $G$ as $$ \mathbb{C}[G]:=\left\{ \sum_{x\in G}c_{x}x \colon c_{x}\in \mathbb{C} \right\}. $$ Now, these expressions are just notation, and as I understand, formally it is actually a set of functions $f\colon G \to \mathbb{C}$ with finite support. My first question is: why use this finite sum notation instead of thinking of these expressions as what they are (functions)? And for example, in Representation Theory of Finite Groups by Benjamin Steinberg, for a finite group $G$ the group algebra $L(G)$ is defined as the set of functions $f\colon G \to \mathbb{C}$, this is $\mathbb{C}^{G}$. Then he talks about the regular representation, and he defines $ \mathbb{C}[G]:=\left\{ \sum_{x\in G}c_{x}x \colon c_{x}\in \mathbb{C} \right\} $, but if $L(G) = \mathbb{C}[G]$ why not use $L(G)$ instead? Or are they different? As far as I know, the text does not make this clarification. I guess what I really want to ask is why denote group algebra functions this way? I have seen that an isomorphism is constructed between $\mathbb{C}[G]$ and $\mathbb{C}^{G}$ ($G$ finite), but in that sense, these elements would be thought of as different from each other.

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    $\begingroup$ It's a tradition. It's also a matter of taste. $\endgroup$
    – kabenyuk
    Aug 6 at 7:21
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    $\begingroup$ Would you also prefer to think of polynomials of $\mathbb R[x]$ as finite support functions $\mathbb N\to \mathbb R$? Maybe under some circumstances but I prefer the finite linear combination picture most of the time. $\endgroup$
    – rschwieb
    Aug 6 at 21:33

3 Answers 3

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In my opinion it is very misleading to think of elements of the group algebra as functions with finite support, because it does not suggest the correct functoriality of the construction with respect to group homomorphisms $f : G \to H$. Namely, given such a homomorphism, we get an algebra homomorphism

$$\mathbb{C}[f] : \mathbb{C}[G] \ni \sum c_g g \mapsto \sum c_g f(g) \in \mathbb{C}[H]$$

which is quite straightforward to guess and define using the standard notation. Try writing this in terms of functions! The natural thing to do with functions is to pull back but here you have to push them forward. It's just less intuitive.

The construction $X \mapsto \mathbb{C}[X]$ for a plain set $X$ is called taking the free vector space and is worth getting familiar with for a number of reasons, since it's a basic step in many other constructions and also since it's an easy introduction to the idea of a free functor.

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  • $\begingroup$ And is this the "group ring"? I have never gotten a grip on "algebras" (among other things) though they sound neat. $\endgroup$
    – Noobie
    Aug 6 at 8:29
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    $\begingroup$ The group ring is a special case of a group algebra when you take $\mathbb{Z}$ as the ring of coefficients (instead of for instance $\mathbb{C}$), so it's just $\mathbb{Z}[G]$. Because rings are the same thing as $\mathbb{Z}$-algebras. $\endgroup$ Aug 6 at 8:31
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The finite sum notation is useful because if we write elements of $\mathbb{C}[G]$ as a finite sum, then we can multiply these elements exactly like we are used to multiplying polynomials, except we take the non-commutativity into account. For example, if $x,y \in G$ then we have, in $\mathbb{C}[G]$, $$(1+y)(1+xy) = 1 + xy + y + yxy$$(because of non-commutativity we should not combine the last term into $y^2x$.)

If the two elements of $G$ commute, then they behave exactly like polynomials. For example, if $x$ and $y$ commute, then in $\mathbb{C}[G]$, we have $(x+y)^3 = x^3 + 3 x^2 y + 3xy^2 + y^3$ exactly as usual.

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To add to the previous answers, I would argue that defining the group algebra in terms of functions is pretty common, but not necessary. For instance you might define $\mathbb{C}[G]$ as any $\mathbb{C}$-algebra representing the functor $A\mapsto \operatorname{Hom}(G,A^\times)$.

I think that a general principle in algebra is to not get too attached to a particular way of defining/representing an object. What matters is the structure, and you should use whatever notation/representation is more convenient for you. If at some point thinking about $\mathbb{C}[G]$ as the space of functions $G\to \mathbb{C}$ with convolution product is more useful, then do it! But usually it's more intuitive to use the common "formal combination of elements" point of view.

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