K-theoretic Poirier-Reutenauer bialgebra

TL;DR

This paper introduces a K-theoretic analogue of the Poirier-Reutenauer Hopf algebra using K-Knuth equivalence, providing a new algebraic framework and rederiving key K-theoretic Littlewood-Richardson rules.

Contribution

It develops a novel K-theoretic algebraic structure based on K-Knuth equivalence, extending the Poirier-Reutenauer algebra to K-theory.

Findings

Defines a K-theoretic Poirier-Reutenauer Hopf algebra

Re-derives K-theoretic Littlewood-Richardson rules

Establishes connections with Buch and Samuel's K-Knuth equivalence

Abstract

We use the K-Knuth equivalence of Buch and Samuel to define a K-theoretic analogue of the Poirier-Reutenauer Hopf algebra. As an application, we rederive the K-theoretic Littlewood-Richardson rules of Thomas and Yong and of Buch and Samuel.