Enriched set-valued P-partitions and shifted stable Grothendieck polynomials

We introduce an enriched analogue of Lam and Pylyavskyy's theory of set-valued P-partitions. An an application, we construct a K-theoretic version of Stembridge's Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this algebra is generated by Ikeda and Naruse's shifted stable Grothendieck polynomials. We give the first proof that the natural skew analogues of these power series are also symmetric. A central tool in our constructions is a K-theoretic Hopf algebra of labeled posets, which may be of independent interest. Our results also lead to some new explicit formulas for the involution omega on the ring of symmetric functions.

Automated summary

An enriched analogue of the theory of set-valued P-partitions was introduced, leading to the construction of a K-theoretic version of Stembridge's Hopf algebra. It was shown that the symmetric part of this algebra is generated by specific shifted stable Grothendieck polynomials, and the skew analogues of these power series are also symmetric. The use of a K-theoretic Hopf algebra of labeled posets was a central tool in these constructions, which may have independent interest.