Balanced Labellings and Schubert Polynomials

We studyBalanced labellingsof diagrams representing the inversions in a permutation. These are known to be natural encodings of reduced decompositions of permutationsw∈ Σn,and we show that they also give combinatorial descriptions of both the Stanley symmetric functionsFwand the Schubert polynomial Swassociated withw.Furthermore, they lead to an explicit basis for the Schubert modules introduced by Kraskiewicz and Pragacz.