Quasisymmetric Hopf algebras for K-theory

Stable Grothendieck polynomials are power series obtained as certain limits of polynomials representing the K-theory classes of Schubert varieties in type A Grassmannians. They are K-theoretic generalizations of Schur functions and provide a generating set with positive structure constants for the algebra of symmetric functions. Ikeda and Naruse have described "shifted" analogues of stable Grothendieck polynomials which correspond in a similar way to K-theory representatives for Schubert varieties in orthogonal and Lagrangian Grassmannians. These families of power series generate two distinct "shifted" subalgebras of symmetric functions. This talk will investigate ways of understanding the properties of these somewhat complicated symmetric functions using combinatorial Hopf algebras. In particular, we will discuss how to realize all stable Grothendieck polynomials and their shifted versions as the images of certain canonical morphisms from relatively simple Hopf algebras of posets to quasisymmetric functions. In the classical setting, this will recover some ideas from work of Lam and Pylyavskyy. All new results are joint work with Joel Lewis.