Title: Combinatorial models for Schubert polynomials,
Abstract: Schubert polynomials are a basis for the polynomial ring that represent
Schubert classes for the flag manifold. In this paper, we introduce and develop
several new combinatorial models for Schubert polynomials that relate them to
other known bases including key polynomials and fundamental slide polynomials.
We unify these and existing models by giving simple bijections between the
combinatorial objects indexing each. In particular, we give a simple bijective
proof that the balanced tableaux of Edelman and Greene enumerate reduced
expressions and a direct combinatorial proof of Kohnert's algorithm for
computing Schubert polynomials. Further, we generalize the insertion algorithm
of Edelman and Greene to give a bijection between reduced expressions and pairs
of tableaux of the same key diagram shape and use this to give a simple
formula, directly in terms of reduced expressions, for the key polynomial
expansion of a Schubert polynomial.