Involution pipe dreams

Involution Schubert polynomials represent cohomology classes of K-orbits in the complete flag variety, where K is the orthogonal or symplectic group. We show that they also represent T-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables xi + xj, and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting.