A Bijective Proof of the Hook-content Formula

Abstract

A bijective proof of the product formula for the principal specialization of super Schur functions (also called hook Schur functions) is given using the combinatorial description of super Schur functions in terms of certain tableaux due to Berele and Regev. Our bijective proof is based on the Hillman–Grassl algorithm and a modified version of Schützenberger's jeu de taquin. We then explore the relationship between our modified jeu de taquin and a modified jeu de taquin by Goulden and Greene. We define a common extension and prove an invariance property for it, thus discovering that both modified jeu de taquins are, though different, equivalent.