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Given a group $G$. Let $\mathbb{C}$ be the complex field. Then $\mathbb{C}G$ is the set of linear combinations of elements of $G$. Addition and multiplication are defined as usual. We can also think $\mathbb{C}G$ as the set of all functions from $G$ to $\mathbb{C}$. Addition is the addition of functions and multiplication of two functions is their convolution.

Now if we are give an algebraic group $G$ and $N$ is a subgroup of $G$. I saw the notation in many places. For example, cluster algebras and representation theory. Why $\mathbb{C}[N]$ is called coordinate ring of $N$? What are the elements of $\mathbb{C}[N]$? What are the additions and multiplications in $\mathbb{C}[N]$. Thank you very much.

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    $\begingroup$ nitpick :In my understanding, $\mathbb CG$ is the set of all functions from G to $\mathbb C$ with finite support $\endgroup$ – gary Aug 6 '11 at 3:33
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In that paper $\mathbb C[N]$ denotes the ring of regular functions (also known as the coordinate ring) on the subgroup $N\subset G$, the maximal unipotent subgroup. $N$ is an affine variety, so this makes sense.

If you do not know what "coordinate ring" means, you should go read some introduction to algebraic geometry (I would suggest the wonderful Ideals, varieties and algorithms by David Cox, John Little and Donal O'Shea, or if you are only interested in algebraic groups, maybe T. A. Springer's Linear Algebraic Groups which is written so as to be (more or less) self-contained—YMMV, of course!), for otherwise it is most likely that Leclerc's paper is going to be pretty much incomprehensible...

You should keep in mind that «we can also think $\mathbb CG$ as the set of all functions from G to $\mathbb C$» only when the group $G$ is finite (and I would add that while you may think that you probably shouldn't) In Leclerc's context this is quite not the case.

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