How to Compute the coordinate ring of flag variety?

Question

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

Answer 1

$G/B$ 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.

Answer 2

The ring $\bigoplus R_{\lambda}$ is generally not a polynomial ring. For example, when $G=SL_3$, this ring is $$\mathbb{C}[p_1, p_2, p_3, p_{12}, p_{13}, p_{23}]/(p_1 p_{23} - p_{13} p_2 + p_{3} p_{12}).$$

In the case of $SL_2$, this is simple enough. The group $SL_2$ acts on the vector space $V$ spanned by $x$ and $y$ in the obvious way. The homogenous degree $n$ polynomials are naturally $\mathrm{Sym}^n \ V$ and thus acquire an action of $SL_2$ as well. One sees that this is the irreducible representation of degree $n+1$.

Answer 3

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.