# Is there an atlas of Algebraic Groups and corresponding Coordinate rings?

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.

• This might help groupprops.subwiki.org/wiki/Main_Page – Jonathan Fischoff Jul 31 '10 at 23:43
• 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]? – Grigory M Aug 1 '10 at 5:11
• 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"? – Pete L. Clark Aug 1 '10 at 11:40
• @Grigory & Pete, the question was crappy, I have hopefully made it more clear. :/ I apologize for being opaque. – BBischof Aug 1 '10 at 17:36
• @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. – BBischof Aug 1 '10 at 17:36

You probably mean for $G$ to be a reductive group. Keep in mind that $G/B$ is equal to $\text{Proj}(R)$ for many different $R$'s, corresponding to different embeddings of $G/B$ into projective space. The best object to study is the homogeneous coordinate ring (also known as the Cox ring) of $G/B$. In that case, when $G = SL_n$, the homogeneous coordinate ring is in Miller and Sturmfels' Combinatorial Commutative Algebra Chapter 14. For the general case, some keywords to look for are "standard monomial theory", "straightening laws", and "Littelmann path model". The homogeneous coordinate ring of a general $G/B$ (or at least $G/P$ for $P$ a maximal parabolic) might be in Lakshmibai and Raghavan's Standard Monomial Theory: Invariant Theoretic Approach, but I am not sure. Regardless, that is a good introduction to the subject and should have a fairly comprehensive list of references for further information.