Let X be a "Bruhat cell" inside the space $M_{kn}$ of $k\times n$ matrices, i.e. an orbit under downward row operations and rightward column operations. If we take its closure, we get a (generalized) matrix Schubert variety. Ezra Miller and I showed how to degenerate X to a union of coordinate spaces, one for each "pipe dream" for X's associated permutation. In particular, X's equivariant cohomology class (the double Schubert polynomial) is controlled by the algebra of divided difference operators, with the effect that no two pipes are allowed to cross twice.

But now take X's conormal bundle first, inside $M_{kn} \times M_{kn}^*$, before taking the closure. Paul Zinn-Justin and I show that if one does the corresponding degeneration, the components are indexed by "Temperley-Lieb pipe dreams" (assuming a natural condition on the permutation). In these, pipes never cross at all, but loops are now possible, each contributing a factor of 2 to the nonreducedness of a component.