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Séminaire Lotharingien de Combinatoire, 82B.63 (2019), 12 pp.

# Zachary Hamaker, Eric Marberg, and Brendan Pawlowski

# Involution pipe dreams

**Abstract.**
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 *x*_{i} + *x*_{j}, 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.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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