Nonsymmetric Schur Functions and an Analogue of the Robinson-schensted-knuth Algorithm

We exhibit a weight-preserving bijection between semi-standard Young tableaux and skyline augmented fillings to provide the first combinatorial proof that the Schur functions decompose into nonsymmetric functions indexed by compositions. The insertion procedure involved in the proof leads to an analogue of the Robinson-SchenstedKnuth Algorithm for skyline augmented fillings. We also prove that the nonsymmetric Schur functions are equal to the standard bases for Schubert polynomials introduced by Lascoux and Schützenberger. This provides a non-inductive construction of the standard bases and a simple formula for the right key of a semi-standard Young tableau.