Abelian groups and representations of finite partially ordered sets
type: Item
instance of: scholarly article
title: Abelian groups and representations of finite partially ordered sets
zbMATH Open document ID: 0959.16011
author: David M. Arnold
published in: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC
publication date: 2000-08-13T00:00:00Z
review text: A representation of a finite partially ordered set (poset) \(S\) over a field \(k\) is an object \(U=(U_0,U_i:i\in S)\), where \(U_0\) is a finite-dimensional vector space over \(k\) and each \(U_i\) is a subspace such that \(U_i\subseteq U_j\) whenever \(i\leq j\) in \(S\). The set of all such objects forms a category \(\text{rep}(S,k)\), where homomorphisms between representations of \(S\) are vector space homomorphisms \(f\colon U_0\to V_0\) that preserve the distinguished subspaces: \(f(U_i)\subseteq V_i\) for all \(i\in S\). The first chapter of the book provides an easily accessible and therefore much welcome introduction to the theory of representations of posets. The study of representations arose out of the study of ``matrix problems'' such as finding canonical forms or showing two matrices are equivalent. These connections are discussed in the first two sections of the book, beginning with the case when \(S\) is an antichain -- a set with no order relations between the elements. The other main topic of the first chapter is representation type. The category \(\text{rep}(S,k)\) has finite representation type if there are only finitely many indecomposable representations of \(S\) up to isomorphism. In 1975, Kleiner showed that \(\text{rep}(S,k)\) has finite representation type if and only if \(S\) does not contain one of five special posets. Infinite representation type is usually allocated into two disjoint classes, tame and wild. Intuitively, a category that doesn't have finite representation type has tame representation type if its indecomposables can be classified, and wild representation type otherwise. In close analogy to the classification of finite type, \(\text{rep}(S,k)\) has tame representation type if and only if \(S\) does not contain any one of six indicator posets as a subposet. Beginning an effective switching-of-settings pattern that continues throughout the book, Chapter 2 offers an introduction to the theory of torsion-free Abelian groups o…
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20K40
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 1489458
zbMATH Keywords: almost completely decomposable groups
zbMATH Keywords: finite partially ordered sets
zbMATH Keywords: representations of posets
zbMATH Keywords: finite representation type
zbMATH Keywords: indecomposable representations
zbMATH Keywords: tame representation type
zbMATH Keywords: wild representation type
zbMATH Keywords: torsion-free Abelian groups of finite rank
zbMATH Keywords: category equivalences
zbMATH Keywords: categories of Butler groups
zbMATH Keywords: quasi-isomorphisms
zbMATH Keywords: near isomorphisms
zbMATH Keywords: endomorphism rings
zbMATH Keywords: direct sums of rank-1 groups
zbMATH Keywords: pure subgroups
zbMATH Keywords: typesets
zbMATH Keywords: quasi-homomorphisms
zbMATH Keywords: almost split sequences
zbMATH Keywords: bracket groups
reviewed by: Charles Vinsonhaler
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1489458