Combinatorial Hopf algebras and K-homology of Grassmanians

TL;DR

This paper explores six combinatorial Hopf algebras related to the K-theory of Grassmannians, introducing new symmetric functions and combinatorial structures inspired by set-valued tableaux.

Contribution

It develops K-theoretic analogues of classical Hopf algebras and introduces new families of symmetric functions based on set-valued combinatorial objects.

Findings

Defined six new combinatorial Hopf algebras.

Introduced a theory of set-valued P-partitions.

Studied three new symmetric functions related to plane partitions and tableaux.

Abstract

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic analogues of the by now classical ``square'' of Hopf algebras consisting of symmetric functions, quasisymmetric functions, noncommutative symmetric functions and the Malvenuto-Reutenauer Hopf algebra of permutations. In addition, we develop a theory of set-valued P-partitions and study three new families of symmetric functions which are weight generating functions of reverse plane partitions, weak set-valued tableaux and valued-set tableaux.