Involution pipe dreams

Abstract

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 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.

arXiv comment36 pages, 2 figures; v2: reordered Sections 4 and 5, added references and examples; v3: some corrections, added exposition; v4: minor corrections; v5: fixed typo in equation (6.3), final version

References in corpus (9)

• Involution words II: braid relations and atomic structures
• Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures
• Polynomials for symmetric orbit closures in the flag variety
• Transition formulas for involution Schubert polynomials
• K-theory formulas for orthogonal and symplectic orbit closures
• On some properties of symplectic Grothendieck polynomials
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Cited by in corpus (2)

• Gröbner geometry for skew-symmetric matrix Schubert varieties
• Crystals for shifted key polynomials