Classical double Grothendieck transitions
Eric Marberg
TL;DR
The paper develops a unified framework for classical-type double Grothendieck transitions, extending type A results to B, C, and D by leveraging Kirillov–Naruse polynomials and Lenart–Postnikov Chevalley formulas. It establishes a finite, positive expansion of type B/C/D K-Stanley functions into K-theoretic Schur P- and Q-functions, and proves that GP and GQ form (sub)bialgebras with positivity of both product and coproduct structure constants. The core tool is a transition operator calculus that expresses Kirillov–Naruse polynomials 𝔊^X_w in terms of 𝔊^X_v, providing recursive β-recurrences for K-theoretic invariants and enabling termination to Grassmannian cases. These results lead to concrete positivity statements for skew P/Q functions and advance conjectures in AHHP regarding C-type expansions, highlighting the deep connections between equivariant K-theory, symmetric functions, and shifted tableaux combinatorics.
Abstract
Kirillov and Naruse have constructed double Grothendieck polynomials to represent the equivariant K-theory classes of Schubert varieties in the complete flag manifolds of types B, C, and D. We derive a recursive formula for these polynomials, extending certain K-theoretic transition equations known in type A to all classical types. As an application, we obtain an identity that expands the K-Stanley symmetric functions in types B, C, and D into positive linear combinations of K-theoretic Schur P- and Q-functions. We also resolve several positivity conjectures related to the skew generalizations of the latter functions.