Schur P-positivity and involution Stanley symmetric functions

TL;DR

This paper proves that involution Stanley symmetric functions are Schur P-positive, provides an efficient algorithm for their decomposition, and characterizes when they equal Schur P-functions, with applications to pattern avoidance and Pfaffian formulas.

Contribution

It establishes Schur P-positivity of involution Stanley symmetric functions and introduces an algorithm for their decomposition into Schur P-summands.

Findings

Involution Stanley symmetric functions are Schur P-positive.

An efficient algorithm for decomposition into Schur P-summands is provided.

Characterization of functions equal to Schur P-functions via pattern avoidance.

Abstract

The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words, and are indexed by the involutions in the symmetric group. By construction each $\hat{F}_y$ is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur $P$-positive. We give an algorithm to efficiently compute the decomposition of $\hat{F}_y$ into Schur $P$-summands, and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur $P$-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur $P$-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila-Serrano and DeWitt on skew Schur functions which are Schur $P$-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials.