Tableau algorithms defined naturally for pictures
Given a skew tableau \(\chi\), we consider two partial orders: \((\chi, \nwarrow)\) in which there is increase from left to right and down columns and \((\chi, \swarrow)\) in which there is increase from left to right and up columns. Given a bijection between the positions in two skew tableaux, \(f:\chi \to\psi\), \(f\) is a picture if \(f(\chi, \nwarrow)\) is consistent with \((\psi, \swarrow)\) and \(f^{-1} (\psi, \nwarrow)\) is consistent with \((\chi, \swarrow) \). The Littlewood-Richardson rule can be stated in terms of pictures: \(c^\lambda_{\mu\nu}\) is the cardinality of the set of pictures from \(\lambda \backslash \mu\) to \(\nu\). Schützenberger's concept of glissement has a natural extension to pictures in which the glissement is applied to either the domain or the range. The author proves that these two forms of glissement commute, and this in turn simplifies the proofs of the main properties of glissement.
scientific article; zbMATH DE number 935387
scientific article