TITLE: How to Compute the coordinate ring of flag variety?
QUESTION [12 upvotes]: 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
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

REPLY [7 votes]: The ring in question may be identified with the ring of $N$-invariant functions on $G$. (Where $N$ is the unipotent radical of $G$.) 
In type $A$, the group $G$ is $n \times n$ matrices with determinant $1$. A function is $N$ invariant if it is preserved by rightward column operations. (That is, adding $a \times (\mbox{column} \ i)$ to $\mbox{column} \ j$, for $i<j$.) It is obvious that those minors which use the left $k$ columns for some $k$ have this invariance. 
In fact the ring is generated by these minors, which are called Plucker coordinates. The relations between these minors, called Plucker relations, are also extremely classical and well known. A good low level reference to this topic is Chapter 14 of Miller and Sturmfels' book Combinatorial Commutative Algebra.