Shifted combinatorial Hopf algebras from $K$-theory

Abstract

In prior joint work with Lewis, we developed a theory of enriched set-valued $P$-partitions to construct a $K$-theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of $K$-theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new $K$-theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures.

32 pages; v2: minor revisions, removed Conjectures 4.37 and 4.42 which now have proofs elsewhere

References in corpus (5)

• Polynomial realizations of some trialgebras
• Shifted insertion algorithms for primed words
• Key and Lascoux polynomials for symmetric orbit closures
• Expanding K-theoretic Schur Q-functions
• Combinatorial formulas for shifted dual stable Grothendieck polynomials

Cited by in corpus (1)

• Crystals for set-valued decomposition tableaux