Shifted combinatorial Hopf algebras from $K$-theory

TL;DR

This paper introduces shifted combinatorial Hopf algebras within $K$-theory, expanding the algebraic framework of peak functions and Schur $P$- and $Q$-functions with new formulas and structures.

Contribution

It constructs a shifted diagram of $K$-theoretic Hopf algebras, providing new analogues of classical peak algebras and studying their properties and duals.

Findings

New $K$-theoretic analogues of peak algebra

Formulas for product, coproduct, and antipode in these algebras

Connections between shifted Hopf algebras and Schur $P$- and $Q$-functions

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.