Antipode formulas for some combinatorial Hopf algebras

TL;DR

This paper provides explicit combinatorial formulas for the antipode maps in certain K-theoretic combinatorial Hopf algebras, extending the understanding of their algebraic structure.

Contribution

It introduces explicit antipode formulas for K-theoretic analogues of several fundamental combinatorial Hopf algebras, filling a gap in their algebraic characterization.

Findings

Explicit antipode formulas for K-theoretic symmetric functions

Explicit antipode formulas for K-theoretic quasisymmetric functions

Explicit antipode formulas for K-theoretic noncommutative symmetric functions

Abstract

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.