Geodesic
A Decomposition of Schur Functions and an Analogue of the Robinson-Schensted-Knuth Algorithm
Séminaire lotharingien de combinatoire, Tome 57 (2007-2010)

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We exhibit a weight-preserving bijection between semi-standard Young tableaux and semi-skyline augmented fillings to provide a 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-Schensted-Knuth Algorithm for semi-skyline augmented fillings. This procedure commutes with the Robinson-Schensted-Knuth Algorithm, and therefore retains many of its properties.

Comment by Sarah Mason

Sarah Mason adds several remarks, clarifying that the nonsymmetric polynomials denoted Ê\alpha(X;q,t) in the paper are equivalent to Demazure characters, introduced by Demazure, and that the specialization of these polynomials studied in the paper has been investigated by Lascoux and Schü:tzenberger under the name of "standard bases" respectively "Demazure atoms." The relevant references are provided. 
@article{SLC_2007-2010_57_a4,
     author = {Sarah Mason},
     title = {A {Decomposition} of {Schur} {Functions} and an {Analogue} of the {Robinson-Schensted-Knuth} {Algorithm}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {57},
     year = {2007-2010},
     url = {http://geodesic.mathdoc.fr/item/SLC_2007-2010_57_a4/}
}
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Sarah Mason. A Decomposition of Schur Functions and an Analogue of the Robinson-Schensted-Knuth Algorithm. Séminaire lotharingien de combinatoire, Tome 57 (2007-2010). http://geodesic.mathdoc.fr/item/SLC_2007-2010_57_a4/