Fixed-point-free involutions and Schur P-positivity

Abstract

The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representatives ${\hat{\mathfrak{S}}}_z^{\mathtt{FPF}}$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions ${\hat{F}}_z^{\mathtt{FPF}}$, which are stable limits of the polynomials ${\hat{\mathfrak{S}}}_z^{\mathtt{FPF}}$, are Schur $P$-positive. To do so, we construct an analogue of the Lascoux-Schützenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest for ${\hat{\mathfrak{S}}}_z^{\mathtt{FPF}}$ when $z$ is a fixed-point-free version of a Grassmannian permutation. We also classify the fixed-point-free involution Stanley symmetric functions that are single Schur $P$-functions, and show that the decomposition of ${\hat{F}}_z^{\mathtt{FPF}}$ into Schur $P$-functions is unitriangular with respect to dominance order on strict partitions. These results and proofs mirror previous work by the authors related to the orthogonal group action on the type A flag variety.

34 pages, 1 figure. This article was formerly the second half of arXiv:1701.02824; v2: revised introduction, expanded proofs and examples, added index of notation, minor corrections

References in corpus (6)

• The On-Line Encyclopedia of Integer Sequences
• 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
• Transition formulas for involution Schubert polynomials
• 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
• Transition formulas for involution Schubert polynomials
• 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
• Atoms for signed permutations
• Gröbner geometry for skew-symmetric matrix Schubert varieties
• Stanley symmetric functions for signed involutions
• On pattern avoidance in matchings and 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
• Crystals for shifted key polynomials