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
integrablehighest 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

and denote it by $S_{w, \Lambda}$.Schubert variety associated to $w$ and $\Lambda$

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.