On some properties of symplectic Grothendieck polynomials

Abstract

Grothendieck polynomials, introduced by Lascoux and Schützenberger, are certain $K$-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the $K$-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the $K$-theoretic Schur $P$-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain "Grassmannian" orbit closures.

arXiv comment24 pages; v2: minor edits, updated references, final version

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• K-theoretic analogues of factorial Schur P- and Q-functions
• Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures
• Schur P-positivity and involution Stanley symmetric functions
• Fixed-point-free involutions and Schur P-positivity
• Polynomials for symmetric orbit closures in the flag variety
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Cited by in corpus (12)

• K-theory formulas for orthogonal and symplectic orbit closures
• A symplectic refinement of shifted Hecke insertion
• Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
• Shifted insertion algorithms for primed words
• Involution pipe dreams
• Gröbner geometry for skew-symmetric matrix Schubert varieties
• Principal specializations of Schubert polynomials in classical types
• Expanding K-theoretic Schur Q-functions
• Combinatorial formulas for shifted dual stable Grothendieck polynomials
• Shifted combinatorial Hopf algebras from $K$-theory
• Crystals for set-valued decomposition tableaux
• Crystals for shifted key polynomials