Schur P-positivity and involution Stanley symmetric functions

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.

38 pages, 3 figures; v2: some typos corrected, updated references; v3: various corrections and a few new results, article now split into two parts with the second half its own submission; v4: updated references and fixed typos, final version

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• 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
• Reduced word enumeration, complexity, and randomization
• Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
• Shifted insertion algorithms for primed words
• Extending a word property for twisted Coxeter systems
• Bumpless Pipedreams, Reduced Word Tableaux and Stanley Symmetric Functions
• Atoms for signed permutations
• Stanley symmetric functions for signed involutions
• Highest weight crystals for Schur Q-functions
• Candidate for the crystal $B(-\infty )$ for the queer Lie superalgebra
• Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions
• Affine transitions for involution Stanley symmetric functions
• Bialgebras for Stanley symmetric functions
• Crystals for shifted key polynomials
• Schubert polynomial analogues for degenerate involutions