Compressed straight tableaux and a distributive lattice of representations
type: Item
instance of: scholarly article
title: Compressed straight tableaux and a distributive lattice of representations
zbMATH Open document ID: 0963.05134
author: B. D. Taylor
published in: Journal of Combinatorial Theory. Series A
publication date: 2001-06-21T00:00:00Z
review text: The paper under review studies the algebraic and combinatorial properties of tableaux which are row-convex and are more general than the Ferrers diagrams of skew partitions. The first main result of the paper is that the set of polynomials associated with compressed straight tableaux forms a basis for the ring of polynomials \({\mathbb{Z}}[t_{ij}\mid 1\leq i\leq m, j\geq 1]\). This basis involves Gröbner bases ideas and is used in the study of the representations of \(\text{GL}_m({\mathbb{Z}})\) corresponding to row-convex shapes. In particular, the basis enjoys the remarkable property that every ``row-convex'' GL\(_m({\mathbb{Z}})\)-submodule of \({\mathbb{Z}}[t_{ij}]\) has a basis of ``compressed straight'' polynomials. This implies that the lattice generated by row-convex representations under intersection and linear span is distributive. The author provides some ring-theoretic applications and studies the Cohen-Macaulay property for rings generated by elements of a representation of the above lattice.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 1525324
zbMATH Keywords: compressed straight tableaux
zbMATH Keywords: bases of polynomial rings
zbMATH Keywords: Cohen-Macaulay property
zbMATH Keywords: row-convex shapes
zbMATH Keywords: lattice of representations
zbMATH Keywords: row-convex representations
MaRDI profile type: Publication
OpenAlex ID: W2086509271
cites work: Characteristic-free representation theory of the general linear group
cites work: Schur functors and Schur complexes
cites work: Projective resolutions of Weyl modules.
cites work: A new construction in homological algebra.
cites work: On lifting maps between Weyl modules: Can bad shapes be resolved by better shapes?
cites work: Q4317713
cites work: Q3792822
cites work: Rings of Invariants of Tori, Cohen-Macaulay Rings Generated by Monomials, and Polytopes
cites work: Specht series for skew representations of symmetric groups
cites work: Presentations of Schur complexes attached to row convex shapes
cites work: Q3033866
cites work: Standard monomial theory for Bott-Samelson varieties
cites work: Borel-Weil theorem for configuration varieties and Schur modules
cites work: Specht series for column-convex diagrams
cites work: Straightening for standard monomials on Schubert varieties
cites work: Percentage-avoiding, northwest shapes and peelable tableaux
cites work: Q5203133
cites work: Q4001520
cites work: Addendum: Specht modules for column-convex diagrams: Characteristic-free results for Weyl modules
cites work: A sagbi basis for the quantum Grassmannian
cites work: Q4132541
cites work: Algorithms in invariant theory
cites work: Q4861423
cites work: Row-convex tableaux and the combinatorics of initial terms
cites work: A straightening algorithm for row-convex tableaux
cites work: A vanishing theorem for Schur modules
DOI: 10.1006/JCTA.2000.3107
Recommended article: A combinatorial representation for a special class of complete distributive lattices
Recommended article: Complements of tableaux and straightening bideterminants
Recommended article: On the representation of finite distributive lattices
Recommended article: Q3410473
Recommended article: Q4286318
Recommended article: A representation theorem for stratified complete lattices
Recommended article: On a representation of distributive lattices
Recommended article: Some combinatorial aspects of congruence lattice representations
Recommended article: Completely representable lattices
Recommended article: Q5144370
description: scientific article; zbMATH DE number 1525324