Diagram Rules for the Generation of Schubert Polynomials

We prove an elegant combinatorial rule for the generation of Schubert polynomials based on box diagrams, which was conjectured by A. Kohnert. The main tools for the proof are (1) a recursive structure of Schubert polynomials and (2) a partial order on the set of box diagrams. As a byproduct we obtain (combinatorial) proofs for two other rules for the generation of Schubert polynomials based on box diagrams: (1) the more complicated rule of N. Bergeron, and (2) the rule of P. Magyar, which we show to be a simplified Bergeron rule. The well-known fact that the Schubert polynomials associated to Grassmannian permutations are in fact Schur polynomials is derived from Kohnertʹs rule.