ag.algebraic geometry - 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 $itimes j$ submatrix has rank at most $r_{ij}$". These are your polynomials cutting out the matrix Schubert variety, e.g.,
$$x_{11}=0, \ det( x_{ij} )_{1leq i,jleq 3} = 0.$$

(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.