Involution pipe dreams

Zachary Hamaker, Eric Marberg, Brendan Pawlowski
2021 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
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 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
more » ... ind 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.
doi:10.4153/s0008414x21000274 fatcat:7munkmlrhze2xfuvhuvff4mroa