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

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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.

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Notes

  1. A graded bialgebra \(H = \bigoplus _{n\ge 0} H_n\) is connected if the unit and counit maps restrict to inverse isomorphisms \(R \cong H_0\). This condition is not essential, but recurs in many examples and is often convenient to assume. Any graded connected bialgebra is automatically a Hopf algebra, and when defining such objects one just needs to specify the (co)product maps.

  2. For simplicity, we define \({\mathfrak {m}}\Pi \textsf {Sym}_{{\mathbb {Q}}[\beta ]}\) over the polynomial ring \({\mathbb {Q}}[\beta ]\) with rational coefficients, but in fact, it would be sufficient to work with coefficients in \({\mathbb {Z}}[2^{-1}]\) rather than \({\mathbb {Q}}\). As in Corollary 4.17, we will only ever need to divide by powers of the prime 2.

  3. Matsumura’s result concerns certain polynomials \(\mathfrak G_\sigma (x,\xi ) \in {\mathbb {Z}}[\beta ][x_1,x_2,\ldots ,\xi _1,\xi _2,\ldots ]\) indexed by permutations \(w \in S_\infty \). These are related to the power series \(G_w\) by the identity \(G_w = \lim _{m\rightarrow \infty } \mathfrak G_{1^m\times w}(x,0)\), where convergence is in the sense of formal power series.

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Acknowledgements

The first author was partially supported by an ORAU Powe award. The second author was partially supported by Hong Kong RGC Grant ECS 26305218. We are grateful to Zach Hamaker, Hiroshi Naruse, Brendan Pawlowski, and Alex Yong for helpful comments.

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  1. Department of Mathematics, George Washington University, Washington, DC, USA

    Joel Brewster Lewis

  2. Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

    Eric Marberg

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Correspondence to Eric Marberg.

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Lewis, J.B., Marberg, E. Enriched set-valued P-partitions and shifted stable Grothendieck polynomials. Math. Z. 299, 1929–1972 (2021). https://doi.org/10.1007/s00209-021-02751-5

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