Grove polynomials and $K$-theoretic quasisymmetry

TL;DR

This paper introduces grove polynomials, extending forest polynomials, and establishes their duality with quasisymmetric Schubert cells, providing a geometric interpretation for certain quasisymmetric functions.

Contribution

It defines grove polynomials as a set-valued extension and proves their $K$-theoretic duality with quasisymmetric Schubert cells, linking algebraic and geometric structures.

Findings

Grove polynomials are dual to quasisymmetric Schubert cells.

Finite truncations of Lam-Pylyavskyy's functions have geometric $K$-theoretic interpretations.

Establishes a new duality framework in $K$-theory and quasisymmetric functions.

Abstract

We define the grove polynomials, a set-valued extension of forest polynomials. We show that they are $K$-theoretically dual to the quasisymmetric Schubert cells which pave the quasisymmetric flag variety, in the same way that Grothendieck polynomials are dual to Schubert cells in the complete flag variety. As a consequence, the finite truncations of the multi-fundamental quasisymmetric functions of Lam-Pylyavskyy acquire a geometric interpretation as $K$-theoretic representatives of quasisymmetric Schubert cells.