Fixed-point-free involutions and Schur $P$-positivity

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 $\hat{\mathfrak{S}}^{\mathrm{FPF}}_{z}$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions $\hat{F}^{\mathrm{FPF}}_{z}$, which are stable limits of the polynomials $\hat{\mathfrak{S}}^{\mathrm{FPF}}_{z}$, 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 $\hat{\mathfrak{S}}^{\mathrm{FPF}}_{z}$ 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 $\hat{F}^{\mathrm{FPF}}_{z}$ 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.

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B. Pawlowski was partially supported by NSF grant 1148634.

Received 12 July 2017

Published 27 September 2019