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

## TL;DR

This paper develops an enriched theory of set-valued P-partitions, constructs a K-theoretic Hopf algebra of peak quasisymmetric functions, and proves symmetry of shifted stable Grothendieck polynomials, advancing algebraic combinatorics.

## Contribution

It introduces an enriched set-valued P-partition framework and a K-theoretic Hopf algebra, linking shifted stable Grothendieck polynomials to symmetric functions.

## Key findings

- Symmetry of skew shifted stable Grothendieck polynomials proved.
- Construction of a K-theoretic Hopf algebra of labeled posets.
- New explicit formulas for the involution on symmetric functions.

## Abstract

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.