Shifted Hecke insertion and the K-theory of OG(n,2n+1)

TL;DR

This paper develops a new combinatorial framework using shifted Hecke insertion to represent the K-theory of the orthogonal Grassmannian, connecting shifted Grothendieck polynomials and K-theoretic structure coefficients.

Contribution

It introduces shifted K-theoretic jeu de taquin and Poirier-Reutenauer algebra to compute K-theory structure coefficients for OG(n,2n+1).

Findings

Constructed symmetric function representatives for K-theory of OG(n,2n+1)

Connected shifted Grothendieck polynomials with K-theory

Recovered structure coefficients using shifted K-theoretic algebra

Abstract

Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We use shifted Hecke insertion to construct symmetric function representatives for the K-theory of the orthogonal Grassmannian. These representatives are closely related to the shifted Grothendieck polynomials of Ikeda and Naruse. We then recover the K-theory structure coefficients of Clifford-Thomas-Yong/Buch-Samuel by introducing a shifted K-theoretic Poirier-Reutenauer algebra. Our proofs depend on the theory of shifted K-theoretic jeu de taquin and the weak K-Knuth relations.