Explicit equations for Schubert varieties

Question

How can one compute the Schubert variety (by compute I mean having actual polynomials that define it) for SL(n)? If this is well known forgive my ignorance and just point me to the right book/paper.

EDIT: Sorry I did not return here for quite some time. It is kind of amusing that the way I learned about Schubert varieties is not even mentioned. Here is how I learned it:

1. $G$ an algebraic group with Lie algebra $\mathbf{g}$.
2. $L(\Lambda)$ is an integrable highest weight module for $\mathbf{g}$.
3. For $w$, an element of the Weyl group, consider the 1-dimensional root space $L(\Lambda )_{w \cdot \Lambda}$.
4. Denote the vector space $U(\mathbf{b}) \bullet L(\Lambda)_{w \cdot \Lambda}$ by $E_w(\Lambda)$ (you take the 1-dimensional root space and act on it by all the raising operators plus the Cartan). Then $E_w(\Lambda) \subset L(\Lambda)$.

Now we are ready: since $L(\Lambda )_{w \cdot \Lambda}$ is 1-dimensional it becomes a single point in $\mathbf{P} \left (E_w(\Lambda) \right)$. We look at the orbit $B \bullet L(\Lambda)_{w \cdot \Lambda} \subset \mathbf{P} \left (E_w(\Lambda) \right)$. We call its closure the Schubert variety associated to $w$ and $\Lambda$ and denote it by $S_{w, \Lambda}$.

I don't know if this is a good way of computing things but in principle it should give you any Schubert variety you need.

Answer 1

If one is learning about this, computing directly with matrices seems like the easiest way (though not as powerful as standard monomials and the toric degenerations that result). Alex Woo's references are a good source for this point of view; I'd also add the first couple chapters of the book Schubert varieties and degeneracy loci by Fulton and Pragacz. And a quick example, in that spirit: to find equations for $X_{2143}$ inside $SL_4/B$, do the following:

(1) Form the rank matrix $(r_{ij})$ for the permutation: $$\left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 \\\ 0 & 0 & 1 & 0 \end{array}\right] \to \left[\begin{array}{cccc} 0 & 1 & 1 & 1 \\\ 1 & 2 & 2 & 2 \\\ 1 & 2 & 2 & 3 \\\ 1 & 2 & 3 & 4 \end{array}\right].$$

(2) Write down the equations on the $4 \times 4$ generic matrix that say "upper-left $i\times j$ submatrix has rank at most $r_{ij}$". These are your polynomials cutting out the matrix Schubert variety, e.g., $$x_{11}=0, \\ \det( x_{ij} )_{1\leq i,j\leq 3} = 0.$$

(3) If you want equations in an open affine patch, set appropriate $x_{ij}$'s to $0$ or $1$, e.g., the opposite cell would have free variables in the $*$ positions: $$\left[\begin{array}{cccc} * & * & * & 1 \\\ * & * & 1 & 0 \\\ * & 1 & 0 & 0 \\\ 1 & 0 & 0 & 0 \end{array}\right].$$

(4) If you want equations in the Plucker embedding, the determinants are built-in to the definition of Plucker coordinates, and you just intersect with appropriate linear subspaces.

In general, these equations are highly redundant, and a big part of the work of Lakshmibai et al, Fulton, and Woo-Yong (as I understand) is to find minimal sets of equations. For the matrix Schubert varietes for $SL_n/B$, the Fulton paper cited by Alex gives a simple answer.

Answer 2

Other possible answers:

1) You might want equations for the preimage of the Schubert variety in GL_n (i.e. the Schubert variety in Stiefel coordinates). Their closures in M_n are matrix Schubert varieties, and their equations are given by Fulton in a paper in Duke Math J. in 1992 (not entirely sure about the year). Knutson and Miller say a lot more about these equations in their Annals of Math. paper (around 2003).

2) You might want equations for local affine neighborhoods in the Schubert variety. This is easy to get from (1) and the description of opposite Schubert cells as sets of matrices. Alex Yong and I work it out in a recent J. Algebra paper. One can derive a different (but somehow equivalent) set of local equations from standard monomial theory also.

Answer 3

Dave Anderson mentions the following basic fact, on which I want to elaborate: Let $V$ be an irreducible representation of the reductive group $G$, and let $G/P$ be the orbit of the high weight vector in $\mathbb{P}(V)$. Let $X=BwP/P$ be any Schubert variety in $G/P$. (For example, $V=\bigwedge^k \mathbb{C}^n$, $G/P$ is the Grassmannian $G(k,n)$ and $X$ is a Schubert variety in the Grassmannian.)

Then the homogenous ideal of $X$ is generated by the homogenous ideal of $G/P$ plus some linear forms. In other words, all you have to understand is (1) the equations of $G/P$ and (2) which elements of $V^*$ vanish on $X$. This result is due to Ramanathan, "Equations defining Schubert varieties and Frobenius splitting of diagonals", Pub. IHES 65 (1987) 61-90.

The equations defining $G/P$ are quadratic. In the case of Grassmannians in the smallest (Plucker) embedding, they are very classical and you can read about them in many places; I like Miller and Sturmfels book Combinatorial Commutative Algebra.

For Grassmanians in the Plucker embedding, it is also easy to describe the linear forms that vanish on a given Schubert variety; they will be the Plucker variables indexed by partitions which do not contain your partition. The general case isn't much worse, but I can't give an answer without first taking time to set up a notations for bases of $V$.

Answer 4

I assume you mean a Schubert variety $X_w=\overline{BwB/B}$ in the flag variety $G/B=X_{w_0}$ in the the Plücker embedding of the latter.

A description of the coordinate ring (and hence of the defining equations) follows from the theory of standard monomials due to Lakshmibai, Musili, Sheshadri. In the case of $SL_n$ and for particular $w\in W=S_n$ everything can be made explicit via combinatorial parametrizations, but there is inherent complexity in the answer — I don't know if I've seen a clean statement for a general $w.$ It becomes more manageable after a toric degeneration.

Answer 5

For the special case of Schubert varieties in the Grassmannian (with respect to the Plücker embedding) you can even get a ready made list of equations from Macaulay 2: link.

The index set for Schubert varieties in $Gr(k,n)$ is the $S_n / (S_k \times S_{n-k})$, so increasing sequences rather than partitions inside of a box.

Answer 6

As the other answers suggest, your question may need better focus to have a helpful answer. Schubert varieties live in a (projective) flag variety as closed subsets and can in principle be defined by homogeneous polynomial equations. They originate however as closures of innocuous affine varieties (Bruhat cells in SL$_n$ or other reductive group) which are isomorphic to affine spaces of dimensions equal to lengths of associated Weyl group elements.

I think the extra difficulty in studying Schubert varieties is that they tend to be singular, which has motivated the use of sheaf cohomology and other indirect tools going beyond a description by polynomials: this was initiated in a lot of concrete detail in an old series of papers at the Tata Institute by Seshadri, Lakshmibai, Musili, as already pointed out. Their focus was especially on the cohomology groups of line bundles. (This is still a tough question in prime characteristic.) Currently Lakshmibai teaches at Northeastern University in Boston and continues to be active in the subject. Other especially active people include Allen Knutson.

For some updated ideas about singularities, the book by Billey and Lakshmibai would be a good source -- though it goes well beyond the sometimes misleadingly simple example of the special linear group:

MR1782635 (2001j:14065) 14M15 (14L35 20F55 20G05), Billey, Sara (1-MIT); Lakshmibai,V. (1-NORE), Singular loci of Schubert varieties. Progress in Mathematics, 182. Birkhauser Boston, Inc., Boston, MA, 2000.