# Partial (or complete) flag varieties as GIT quotients of affine spaces

I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would like to find examples of an affine variety space $V/\mathbb{C}$, a reductive group G acting on V, and a linearization $L$ such that the GIT quotient $V//G$ is equal to the flag variety $Fl(d_1,d_2,\dots,d_k)$.

Of course $\mathbb{P}^{n-1}$ is an easy example for $Fl(1,n)$, where $V = \mathbb{C}^{n}$ and $G = \mathbb{C}^{\times}$. And the Grassmannian $Fl(k,n) = Gr(k, {n})$ can be constructed as a GIT quotient of the vector space $M_{k,{n}}$ of $k\times n$ matrices $Gr(k, {n}) = M_{k,{n}}//GL(k)$, where $GL(k)$ acts as matrix multiplication on the left.

Are there constructions that work for more general choices of $d_i$? Does anyone know of any other examples or have recommendations of where I might look for them?

• Did I misread the question? Do you mean as quotients of affine varieties (which as Jason points out is very easy to arrange), or as quotients of affine spaces (as in your examples)? Aug 21, 2013 at 13:51
• Sorry--I meant affine spaces, although Jason's answer about Mori dream spaces was new to me and very interesting. Aug 21, 2013 at 14:10

If you're willing to quotient by a nonreductive group, then $M_n//B$ will get you the $GL(n)$ flag manifold. (People are usually afraid to do so, worrying that the ring of invariants won't be Noetherian, but this one is.)

That flag manifold is also available reductively. Let $V_0,V_1\ldots,V_n$ be a list of vector spaces with those dimensions, and let $Hom := \prod_{i=1}^n Hom(V_{i-1},V_i)$. If we quotient this by $GL(V_1)\times \cdots \times GL(V_{n-1})$, it forgets the actual maps and only remembers the images inside $V_n$, so the result is (or to be precise, can be chosen to be) the manifold of flags in $V_n$. I forget whom this is due to, but it's pretty old.

You can get some of this to work for symplectic and orthogonal groups, using the $O(V) \times Sp(W)$ action on $V\otimes W$; the reference I know is [Lerman-Montgomery-Sjamaar].

• Both of Allen's tricks also get you partial flag manifolds in type $A$, replacing $B$ by $P$ in the first case, and using only the $V_i$ of the dimensions we care about in the second. Aug 21, 2013 at 16:20

Short answer: Every spherical variety is a Mori Dream Space.

Longer answer: Every projective variety is a "GIT quotient", but the most obvious construction is probably unenlightening. Let $X$ be a projective variety embedded in $\mathbb{P}^n$. Let $V$ be the affine cone over $X$ with the natural action of $G=\mathbb{G}_m$. Then $V$ is a $G$-invariant, Zariski closed subset of the affine space $\mathbb{A}^{n+1}$ that has the "standard" linear representation of $G$ (by scaling), and the GIT quotient $V//G$ is $X$.

However, the construction above depends on the choice of a projective embedding, or at least of an ample invertible sheaf, so it is not canonical. There is a class of projective varieties that are canonically GIT quotients, namely the Mori Dream Spaces first studied by Hu and Keel. Every flag variety, and indeed every projective variety homogeneous under a linear algebraic group, is a Mori Dream Space. In fact, there is a class of varieties that contains both projective homogeneous varieties and toric varieties (another large class of Mori Dream Spaces), namely "spherical varieties". Every spherical variety is a Mori Dream Space.

Edit. The OP clarified that he wants $V$ to be an affine space, not an affine variety. For most Mori Dream Spaces, $V$ is not an affine space (for toric varieties, $V$ is an affine space, as follows from the theory of the Cox ring).

• In fact, in the construction of a Mori Dream Space $X$ as a GIT quotient $V // G$ via the Cox ring, $V$ is an affine space if and only if $X$ is a toric variety. Aug 21, 2013 at 15:18