Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ring $ \mathbb{C}[G/B]$, it is known that it is a graded polynomial ring quotient by some homogeneous ideal. For example, consider flag variety of $sl\_2$, it is $\mathbb{P}^{1}$, so the coordinate ring is $\mathbb{C}[x\_1,x\_2]$ as graded ring.


I want to know for general case, i.e. flag variety of finite dimensional Lie algebra $g$($G/B$), how to compute $\mathbb{C}[G/B]$? But we know that from Borel-Weil, we can write it as


$\bigoplus\_{\lambda\in P\_+}$ $R\_\lambda$, where $R\_\lambda$ is irreducible highest dominant weight representations of $g$


My question
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How to compute $\mathbb{C}[G/B]$? explicitly?


How to build explicit ring isomorphism from $\bigoplus\_{\lambda\in P\_+}$ $R\_\lambda$ to some concrete ring?


This question might be elementary. Thanks anonymous and Ben pointing out the stupid mistake I made



How to Compute the coordinate ring of flag variety?
$G/B$ is most naturally a multi-projective variety, embedding in the product of projectivizations of fundamental representations: $\prod \mathbb{P}
(R\_{\omega\_i})$. So there is a multi-homogeneous coordinated ring on $G/B$. You mentioned that this ring is $\bigoplus\_{\lambda\in P\_+}$ $R\_\lambda$. This is correct, and the grading is also apparent. It's given by the weight lattice. (To be more canonical, you should take the duals of every highest weight representation, but what you've written down is isomorphic to that.)


So all that remains is giving the multiplication law on $\bigoplus\_{\lambda\in P\_+}$ $R\_\lambda$. You need to specify maps $R\_\lambda \otimes R\_\mu \mapsto R\_{\lambda+\mu}$. There's a natural candidate: If you decompose $R\_\lambda \otimes R\_\mu$ into a direct sum of irreducible representations, $R\_{\lambda+\mu}$ will appear exactly once. The multiplication law is simply projection onto this factor.


@Shizuo: For $sl\_n$, the situation is more explicit. The flag variety here is the set of flags $0 = V\_0 \subset V\_1 \subset \cdots \subset V\_{n-1} \subset V\_n = \mathbb{C}^n$ with $\dim V\_i = i$. So the flag variety is a closed subvariety of the product of Grassmannians $Gr(1,n) \times \cdots \times Gr(n,n) $. Each of these Grassmannians have a explicitly Plücker embedding into the projectivization of the exterior power of $\mathbb{C}^n$. In particular, the homogeneous ideal is explicitly given by the Plücker relations. So the multi-homogeneous coordinate ring of $Gr(1,n) \times \cdots \times Gr(n,n) $ is just the tensor product of the known homogeneous coordinate rings.


Finally, to get the multi-homogeneous coordinate ring for the flag variety, we need to specify an incidence locus inside $Gr(1,n) \times \cdots \times Gr(n,n) $. Namely, we need to specify those tuples of subspaces that form a flag. But this is easy: it corresponds to certain wedge products being zero. Just impose those additional relations, i.e. mod out by the corresponding multi-homogeneous ideal. Now you should have an explicit, albeit fairly long description of the homogeneous coordinate ring. The above answer for $SL\_3$ looks like a special case of this construction.


