Transition formulas for involution Schubert polynomials

Abstract

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 ${\hat{\mathfrak{S}}}_y^{\mathtt{FPF}}$ (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product of ${\hat{\mathfrak{S}}}_y$ (respectively, ${\hat{\mathfrak{S}}}_y^{\mathtt{FPF}}$) 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 ${\hat{\mathfrak{S}}}_y^{\mathtt{FPF}}$ 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.

31 pages; v2: updated references and acknowledgments; v3: added references, minor corrections; v4: a few more references, examples, and corrections, final version

References in corpus (5)

• Involution words II: braid relations and atomic structures
• Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures
• Schur P-positivity and involution Stanley symmetric functions
• Fixed-point-free involutions and Schur P-positivity
• The Orbits of the Symplectic Group on the Flag Manifold

Cited by in corpus (15)

• Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures
• Schur P-positivity and involution Stanley symmetric functions
• Fixed-point-free involutions and Schur P-positivity
• K-theory formulas for orthogonal and symplectic orbit closures
• On some properties of symplectic Grothendieck polynomials
• A symplectic refinement of shifted Hecke insertion
• On some actions of the 0-Hecke monoids of affine symmetric groups
• Bumping operators and insertion algorithms for queer supercrystals
• Atoms for signed permutations
• Involution pipe dreams
• Gröbner geometry for skew-symmetric matrix Schubert varieties
• Stanley symmetric functions for signed involutions
• Affine transitions for involution Stanley symmetric functions
• Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions
• Schubert polynomial analogues for degenerate involutions