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I was wondering if there was a resource that listed known algebraic groups and their corresponding coordinate rings.

Edit: The previous wording was terrible.

Given an algebraic group $G$, with Borel subgroup $B$ we can form the Flag Variety $G/B$ which is projective. I am hoping for a list of the graded ring $R$ such that $Proj(R)$ corresponds to this Flag Variety.

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    This might help http://groupprops.subwiki.org/wiki/Main_Page2010-07-31
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    I don't quite understand the question. For example, there are a lot of abelian varieties -- what should be listed for them? And in what sense SL_2 (3-dimensional group) corresponds to k[x_0,x_1]?2010-08-01
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    It is a standard exercise to write down the coordinate ring of $GL_n$ as a hypersurface in affine $n^2+1$-space. After doing that, every linear algebraic group is a closed subgroup of $GL_n$, usually given by explicit polynomial equations, so this is easily done. What do you mean by "the projectivizations"?2010-08-01
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    @Grigory & Pete, the question was crappy, I have hopefully made it more clear. :/ I apologize for being opaque.2010-08-01
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    @Jonathan, thanks for the link, an initial exploration has not yielded what I am looking for, but that does not mean it does not exist.2010-08-01
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    http://mathoverflow.net/questions/23426/how-to-compute-the-coordinate-ring-of-flag-variety maybe?2010-08-01
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    @Grigory Yes I know of the question, and the I know the asker personally. I know how to compute them in simple cases(like those he mentions), I was hoping for resources that list answers.2010-08-01
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    @BBischof, I am not sure what the form of the ideal answer you expect will be. Obviously one can "list" such things using Dynkin diagrams (plus a choice of positive weight, corresponding to a projective embedding), since the complete flag variety $G/B$ depends only on the Dynkin diagram of $G$. But presumably you want a list that contains more information. So, you want generators and relations? A homogeneous basis for each such graded ring, together with a rule for multiplication? It's not clear to me yet.2012-08-16

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