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Transition formulas for involution Schubert polynomials. (English) Zbl 06941774
Summary: 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 \(S_n\). Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as \({\hat{\mathfrak S}}_y\) (to be called involution Schubert polynomials) and \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\) (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product of \({\hat{\mathfrak S}}_y\) (respectively, \(\hat{\mathfrak S}^{\mathtt{FPF}}_y\)) 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 \({\hat{\mathfrak S}}_y\) and \( \hat{\mathfrak S}^{\mathtt{FPF}}_y\) appearing in the literature. Our formulas also imply combinatorial identities about involution words, certain variations of reduced words for involutions in \(S_n\). 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 \(S_n\) restricted to involutions.

MSC:
20B30 Symmetric groups
14M15 Grassmannians, Schubert varieties, flag manifolds
05E05 Symmetric functions and generalizations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
14M27 Compactifications; symmetric and spherical varieties
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