Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras

TL;DR

This paper proposes viewing affine Grassmannian Schubert classes as Schur-positive symmetric functions, providing geometric insights into the Schur positivity of k-Schur functions and connecting this to combinatorial Hopf algebra theory.

Contribution

It introduces a geometric perspective on affine Schubert classes as Schur-positive functions and links this to the framework of combinatorial Hopf algebras.

Findings

Geometric explanation for Schur positivity of k-Schur functions

Connection established between affine Schubert classes and symmetric functions

Integration of Schubert calculus with combinatorial Hopf algebra theory

Abstract

We suggest the point of view that the Schubert classes of the affine Grassmannian of a simple algebraic group should be considered as Schur-positive symmetric functions. In particular, we give a geometric explanation of the Schur positivity of k-Schur functions (at t = 1). We also put this in the context of the theory of combinatorial Hopf algebras.