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.



Explicit equations for Schubert varieties
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.


