A combinatorial construction of the Schubert polynomials

Abstract

We show a combinatorial rule based on diagrams (finite nonempty sets of lattice points ( i, j) in the positive quadrant) for the construction of the Schubert polynomials. In the particular case where the Schubert polynomial is a Schur function we give a bijection between our diagrams and column strict tableaux. A different algorithm had been conjectured (and proved in the case of vexillary permutations) by A. Kohnert (Ph.D. dissertation, Universität auf Bayreuth, 1990). We give, at the end of this paper, a sketch of how one would show the equivalence of the two rules.