A symplectic refinement of shifted Hecke insertion

Abstract

Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials $G_{\pi }$ indexed by permutations in the basis of stable Grothendieck polynomials $G_{\lambda }$ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of "orthogonal" and "symplectic" shifted analogues of $G_{\pi }$ in Ikeda and Naruse's basis of $K$-theoretic Schur $P$-functions.

41 pages; v2: fixed several errors, minor reorganization; v3: further corrections, condensed exposition; v4: minor changes; v5: corrected a mistake in Definition 3.5, fixed several typos and minor errors

References in corpus (5)

• K-theory formulas for orthogonal and symplectic orbit closures
• On some properties of symplectic Grothendieck polynomials
• Bumping operators and insertion algorithms for queer supercrystals
• On involutions in symmetric groups and a conjecture of Lusztig
• Queer Supercrystal Structure for Increasing Factorizations of Fixed-Point-Free Involution Words

Cited by in corpus (10)

• On some properties of symplectic Grothendieck polynomials
• K-theory formulas for orthogonal and symplectic orbit closures
• Bumping operators and insertion algorithms for queer supercrystals
• Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
• Shifted insertion algorithms for primed words
• Extending a word property for twisted Coxeter systems
• Gröbner geometry for skew-symmetric matrix Schubert varieties
• Highest weight crystals for Schur Q-functions
• Principal specializations of Schubert polynomials in classical types
• Crystals for shifted key polynomials