Resumen de Transition formulas for involution Schubert polynomials

Zachary Hamaker, Eric Marberg, Brendan Pawlowski

• The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group Sn. Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as Sˆ y (to be called involution Schubert polynomials) and Sˆ FPF y (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product of Sˆ y (respectively, Sˆ FPF y ) with any y-invariant linear polynomial as a linear combination of other involution Schubert polynomials. These identities serve as analogues of Lascoux and Schützenberger’s transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions of Sˆ y and Sˆ FPF y appearing in the literature. Our formulas also imply combinatorial identities about involution words, certain variations of reduced words for involutions in Sn. We construct operators on involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of Sn restricted to involutions.