TITLE: Explicit equations for Schubert varieties
QUESTION [8 upvotes]: 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:


$G$ an algebraic group with Lie algebra $\mathbf{g}$.
$L(\Lambda)$ is an integrable highest weight module for $\mathbf{g}$.
For $w$, an element of the Weyl group, consider the 1-dimensional root space $L(\Lambda )_{w \cdot \Lambda}$.
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.

REPLY [6 votes]: 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 $V$.