A Decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth Algorithm

TL;DR

This paper introduces a combinatorial proof showing Schur functions decompose into nonsymmetric functions using a bijection between Young tableaux and semi-skyline augmented fillings, and develops an RSK analogue that preserves key properties.

Contribution

It provides a new combinatorial decomposition of Schur functions and introduces an RSK-like algorithm for semi-skyline augmented fillings, extending classical combinatorial methods.

Findings

Established a weight-preserving bijection between tableaux types

Proved Schur functions decompose into nonsymmetric functions

Developed an RSK analogue that retains key properties

Abstract

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 RSK algorithm, and therefore retains many of its properties.