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 shift...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2021-12, Vol.299 (3-4), p.1929-1972
Main Authors: Lewis, Joel Brewster, Marberg, Eric
Format: Article
Language:English
Subjects:
Algebra
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Power series
Rings (mathematics)
Set theory
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Summary: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 ω on the ring of symmetric functions.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-021-02751-5