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

## TL;DR

This paper proves that fixed-point-free involution Stanley symmetric functions are Schur P-positive, introduces a recurrence similar to Lascoux-Schützenberger trees, and provides a Pfaffian formula for certain cases, advancing understanding of symmetric functions and geometric representations.

## Contribution

It establishes Schur P-positivity for fixed-point-free involution Stanley symmetric functions and develops an algebraic recurrence analogous to Lascoux-Schützenberger trees.

## Key findings

- Proved Schur P-positivity of fixed-point-free involution Stanley symmetric functions.
- Constructed an algebraic recurrence similar to Lascoux-Schützenberger trees.
- Derived a Pfaffian formula for certain polynomial representatives.

## 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 $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions $\hat{F}^{\tt{FPF}}_z$, which are stable limits of the polynomials $\hat{\mathfrak{S}}^{\tt{FPF}}_z$, are Schur $P$-positive. To do so, we construct an analogue of the Lascoux-Sch\"utzenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest for $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ 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 $\hat{F}^{\tt{FPF}}_z$ 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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Source: https://tomesphere.com/paper/1706.06665