K-theory formulas for orthogonal and symplectic orbit closures

Abstract

The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these orbits. Our polynomials are analogous to the Grothendieck polynomials representing $K$-classes of Schubert varieties, and we show that like Grothendieck polynomials, they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property. We show that the same polynomials represent the equivariant $K$-classes of symmetric and skew-symmetric analogues of Knutson and Miller's matrix Schubert varieties. We derive explicit expressions for these polynomials in special cases, including a Pfaffian formula relying on a more general degeneracy locus formula of Anderson. Finally, we show that taking an appropriate limit of our representatives recovers the $K$-theoretic Schur $Q$-functions of Ikeda and Naruse.

arXiv comment41 pages; v2: fixed some typos, updated references; v3: several minor corrections and clarifications, final version

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Cited by in corpus (12)

• On some properties of symplectic Grothendieck polynomials
• A symplectic refinement of shifted Hecke insertion
• Shifted insertion algorithms for primed words
• Involution pipe dreams
• Gröbner geometry for skew-symmetric matrix Schubert varieties
• Expanding K-theoretic Schur Q-functions
• Combinatorial formulas for shifted dual stable Grothendieck polynomials
• Principal specializations of Schubert polynomials in classical types
• Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties
• Shifted combinatorial Hopf algebras from $K$-theory
• Crystals for set-valued decomposition tableaux
• Crystals for shifted key polynomials