Planar decompositions of tableaux and Schur function determinants
type: Item
instance of: scholarly article
title: Planar decompositions of tableaux and Schur function determinants
zbMATH Open document ID: 0832.05097
DOI: 10.1016/0195-6698(95)90002-0
published in: European Journal of Combinatorics
publication date: 1996-02-08T00:00:00Z
review text: The authors give a general expression for the skew Schur functions as determinant of Schur functions of strip shape, obtained through a planar outside decomposition of the skew shape. This determinantal identity contains the classical Jacobi-Trudi determinant, its dual, the Giambelli determinant and more recent Lascoux and Pragacz's rim ribbon determinant as particular cases. A planar outside decomposition of the shape \(\lambda/\mu\) is a totally ordered \(m\)-tuple \((\theta_1,\dots, \theta_m)\) of strips in the diagram of \(\lambda/\mu\) such that their disjoint union is the shape \(\lambda/\mu\), with the condition that each strip starts on the left or bottom perimeter of the diagram and ends on the right or top perimeter of the diagram. Given two strips \(\theta_i\) and \(\theta_j\) of a diagram, the authors define a new strip \(\theta_i\# \theta_j\), obtained in a norm-commutative way from superimposing \(\theta_i\) and \(\theta_j\). The main result states that, for any fixed outside decomposition \((\theta_1,\dots, \theta_m)\) of \(\lambda/\mu\), the following identity holds: \(s_{\lambda/\mu}(X)= \text{det}(s_{\theta_i\# \theta_j}(X))_{m\times m}\), where \(s_{\lambda/\mu}\) denotes the Schur function associated to \(\lambda/\mu\). The proof is combinatorial and establishes a bijection between the set of all tableaux of a given shape with a fixed outside decomposition, and non-intersecting \(m\)-tuples of lattice paths. The paths used here have four types of permissible steps: up vertical, down vertical, right horizontal and down diagonal. An outside decomposition of a shape induces a decomposition of a tableau of the same shape as a union of tableaux of strip shape. Each tableau of strip shape in the original tableau is bijectively mapped to a lattice path, in which every non-vertical step ending at height \(i\) corresponds to an \(i\) in a box of the tableau of strip shape. The authors verify in detail that an \(m\)-tuple of intersecting lattice paths does…
reviewed by: Claudia Malvenuto
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 798912
zbMATH Keywords: planar decompositions
zbMATH Keywords: Schur function determinants
zbMATH Keywords: symmetric functions
zbMATH Keywords: Young tableaux
zbMATH Keywords: skew Schur functions
zbMATH Keywords: determinantal identity
zbMATH Keywords: determinant
zbMATH Keywords: outside decomposition
zbMATH Keywords: Schur function
zbMATH Keywords: shape
zbMATH Keywords: tableau
zbMATH Keywords: lattice path
zbMATH Keywords: Gessel-Viennot lattice path
author: Angèle. M. Hamel
author: I. P. Goulden
MaRDI profile type: Publication
full work available at URL: 0195-6698(95)90002-0
OpenAlex ID: W1992114298
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cites work: Ribbon Schur functions
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cites work: Nonintersecting paths, pfaffians, and plane partitions
cites work: Lattice path proof of the ribbon determinant formula for Schur functions
description: scientific article; zbMATH DE number 798912