An algorithmic process introduced by A. Kohnert, beginning with specific box diagrams in the first quadrant of the plane, has been used to give manifestly positive combinatorial formulas for certain families of polynomials that are important in geometry and representation theory.

We consider the polynomials arising from applying Kohnert's algorithm to any starting diagram, which we call Kohnert polynomials. We establish some structural results concerning these polynomials, including that their stable limits are quasisymmetric. As an application, we introduce a basis consisting of Kohnert polynomials which stabilises to a new basis of quasisymmetric functions that contains the famous Schur basis of symmetric functions. We additionally give a representation-theoretic interpretation of this new basis. (Includes joint work with S. Assaf.)