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One of the problems in some work I'm doing tells me to consider $GL_2$ acting on $\mathbb{C}[X,Y]$, induced by the natural representation of $GL_2$ on $\mathbb{C}^2$. I just wanted to check 2 things: firstly, is $\mathbb{C}[X,Y]$ typically the set of sums of the form $\sum_{i,j \geq 0}a_{i\,j}X^i Y^j$ $(*)$?

Secondly, what is this action induced by the natural representation on $\mathbb{C}^2$? Obviously in $\mathbb{C}^2$ it's just applying the matrix to the vector; if I have understood $\mathbb{C}[X,Y]$ correctly, is it the action for which the matrix $(A)_{i,j}$ sends $X \longmapsto A_{11}X + A_{12}Y$ and sends $Y \longmapsto A_{21}X+A_{22}Y$? (And then in the sum $(*)$, $X^i \longmapsto (A_{11}X + A_{12}Y)^i$, etc.)

Is that correct? If not, please let me know what I got wrong! Thanks

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That's correct. –  Alex B. Apr 26 '12 at 8:19
    
Just a puntualization. $\mathbb{C}[X,Y]$ is typically the set of finite sums of the form you outlined (or the set of polynomials in two variables over $\mathbb{C}$). Not to be confused with the formal power series ring $\mathbb{C}[[X,Y]]$. –  Giovanni De Gaetano Apr 26 '12 at 8:32
    
$\Bbb{C}[X,Y]$ is a free module with basis $X^iY^j$ so you are correct in the definition of $\Bbb{C}[X,Y]$. –  fpqc Apr 26 '12 at 8:35
    
@Giovanni: Yes, sorry - I meant to say finite! Thank you all for the help. –  Ben Apr 26 '12 at 8:43
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