Diagram Rules for the Generation of Schubert Polynomials

✍ Scribed by Rudolf Winkel

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 332 KB
• Volume: 86
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1998.2931

✦ Synopsis

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 Kohnert's rule.

1999 Academic Press

To every finite permutation ? of natural numbers contained in one of the symmetric groups S n there is associated a Schubert polynomial

such that the collection of all X ? forms a Z-basis of Z[x]. The significance of Schubert polynomials rests mainly on two facts: (1) the ring of Schubert polynomials represents faithfully the ring of cohomology classes of flag manifolds under the cup product, and (2) Schur polynomials are special Schubert polynomials. The theory of Schubert polynomials has been established in a sequence of works by A.