Involution Schubert polynomials represent cohomology classes of K-orbit closures in the complete flag variety, where K is the orthogonal or symplectic group. We show they also represent $\mathsf {T}$ -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 $x_i + x_j$ , and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey–Jockusch–Stanley formula for Schubert polynomials. 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.

对合舒伯特多项式表示完整标志变体中K轨道闭包的上同调类，其中K是正交或辛群。我们表明它们还表示由对称或斜对称矩阵空间中的左上秩条件定义的子变体的 $\mathsf {T}$ -等变上同调类。这种几何意味着这些多项式是变量 $x_i + x_j$ 中单项式的正组合，并且我们给出了这种明确的公式，作为新对象的总和，称为对合白日梦。我们的公式类似于舒伯特多项式的 Billey-Jockusch-Stanley 公式。在 Knutson 和 Miller 的矩阵舒伯特变体的方法中，白日梦公式反映了这些变体理想的 Gröbner 退化，我们推测在我们的环境中识别出类似的退化。