Defining Equations of a Flag Variety

Question

I've been reading Fulton's Young Tableaux, and I'm trying to understand flag varieties. I want to understand the defining equations of a Flag Variety, but the coordinates in Fulton's Plucker relations, and in the equations defining the flag varieties are in the dual projective space. I tried to understand this proof, but I'm very confused.

Does anyone know another source, where they write down the flag's defining equations without using the dual space?

I've searched and searched, but I haven't been able to find anything. Everyone seems to mention it, but they always reference Fulton.

Thanks!

Answer 1

Equations are for cutting schemes out of ambient schemes. Are you sure you want to cut your flag manifold out of projective space? There are easier places to find it.

To begin with, you could embed $Flags(n) \to \prod_{k=1}^{n-1} Gr_k(n)$, taking the flag to its list of subspaces. Then the equations are "for each $i < j$, the $i$-plane should be contained in the $j$-plane".

You could Plücker embed each Grassmannian $Gr_k(n)$ into projective space, or you could regard it as $GL(k) \backslash \backslash M_{k\times n}$, i.e. look at row-spans of $k\times n$ matrices. Then the equations above say that when you stack your $i\times n$ matrix and $j\times n$, the resulting $(i+j)\times n$ matrix should only have rank $j$, so all $(j+1)\times (j+1)$ determinants should vanish. There's some equations.

If you do Plücker embed, it means you only have the Plücker coordinates on those Grassmannians, and so you get Plücker relations between the Plücker coordinates of size $i$ and $j$. I find these much harder to remember than the determinants above.

Once you've Plücker embedded the Grassmannians, then you can Veronese them each by different amounts, then Segre the whole thing together, and you get all the projecively normal embeddings of the flag manifold. It's interesting to note that all the equations encountered along the way are linear or quadratic (a theorem of Ramanathan for general $G$).

Anyway one very good answer to your actual question is [Miller-Sturmfels], chapter 15 I think it is, as Victor Protsak suggested.

Answer 2

One point of view on equations defining flag varieties (and more generally Schubert varieties) is the following, which puts the construction of defining equations in a representation-theoretic context. The following idea works over an arbitrary algebraically closed field, so let's fix an algebraically closed field $k$ of any characteristic. Let $G$ be a semisimple algebraic group over $k$ and denote by $X = G/B$ the flag variety of $G$. For regular dominant $\lambda$ let $V(\lambda)$ denote the Weyl module for $G$ of highest weight $\lambda$. Then we have the standard embedding $ X \hookrightarrow \mathbb P(V(\lambda)) $ and we would like to understand the defining relations for this embedding.

Now, it is known that $X$ is projectively normal under this embedding. So the homogeneous coordinate ring of $X$ is given by $$ R = \bigoplus_{n \geq 0} \Gamma \big( X, \mathcal O(n)|_X \big) = \bigoplus_{n \geq 0} H^0( -n\lambda ) = \bigoplus_{n \geq 0} V(n\lambda)^*, $$ where $H^0(-\lambda)$ is the induced module for $G$ of lowest weight $-\lambda$. On the other hand, we have $$ T := \bigoplus_{n \geq 0} \Gamma \big( \mathbb P(V(\lambda)), \mathcal O(n) \big) = \bigoplus_{n \geq 0} S^n H^0(-\lambda) , $$ the homogeneous coordinate ring of $\mathbb P(V(\lambda))$. So to understand the defining equations of $X$ we need to find the kernel of the restriction morphism $T \twoheadrightarrow R$. This is in general a subtle question! It amounts to understanding for each $n \geq 0$ the kernel of the natural $G$-equivariant morphism $S^n H^0(-\lambda) \twoheadrightarrow H^0(-n\lambda)$ given by multiplication in the ring $R$, which is a nontrivial (but interesting!) representation-theoretic problem.

There are a number of approaches to this problem. I am far from an expert in this area, so I can't give very precise details, but I know of at least two ways to handle this. One is representation-theoretically: since the map $S^n H^0(-\lambda) \twoheadrightarrow H^0(-n\lambda)$ is a $G$-module map one can try to analyze the kernel representation-theoretically. However, if you want some explicit, hands-on formulas, I believe this is where standard monomial theory comes in. In standard monomial theory, a nice basis of these modules is constructed, and I believe you can then explicitly describe this kernel in a combinatorial fashion using that construction. There are by now many papers on standard monomial theory, including some general overview articles, so a Google search should bring up more reading in that direction if you are interested.