Abelian groups and representations of finite partially ordered sets.

*(English)*Zbl 0959.16011
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 2. New York, NY: Springer. xii, 244 p. (2000).

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 of finite rank (henceforth abbreviated tffr groups). The connection between tffr groups and representations is established in Chapter 3, which contains a discussion of Butler groups and a category equivalence between categories of Butler groups and \(Q\)-representations of associated posets. Chapter 2, in the meantime, discusses fundamental tools such as quasi-isomorphism, isomorphism at \(p\) and near isomorphism. There is also a section on stable range conditions and cancellation properties (based on Arnold’s 1981 book), and one on endomorphism rings of tffr groups, including Corner’s Theorem and the Beaumont-Pierce Principal Theorem.

A Butler group is a homomorphic image of a finite direct sum of rank-1 groups, namely subgroups of \(Q\). Butler groups have proved remarkably tractable, in strong contrast to tffr groups in general. They were originally defined and studied by Köhler and Butler in the mid-60’s. Butler, for example, showed that a tffr group is the homomorphic image of a finite direct sum of rank-1 groups if and only if it is a pure subgroup of a finite direct sum of rank ones. Dave Arnold was largely responsible for “popularizing” Butler groups through several papers and his 1981 book. If \(G\) is a Butler group then \(G\) has a finite typeset that is contained in a finite lattice of types \(L\). The category of Butler groups with typeset contained in \(L\) and morphism sets given by \(Q\otimes\text{Hom}(G,H)\), is equivalent to the representations over \(Q\) of the poset \(S\) defined by taking the join-irreducible elements of \(L\) with the opposite ordering imposed. This seminal result allows one to prove theorems about Butler groups and quasi-homomorphisms while working on the familiar ground of vector spaces over \(Q\). Since most results don’t depend on the field, one ends up with much more general theorems in the representation setting. This fact seems to have been under-utilized by the Abelian group community. After a summary of basic results, Chapter 3 concludes with some material on countable Butler groups, including the important characterization that \(G\) is Butler if and only if \(\text{Bext}(G,T)=0\) for all torsion groups \(T\).

In Chapter 4, we switch back to representations, this time over a discrete valuation ring (DVR), with \(Z_p\) being the model. If \(S\) is a poset, \(R\) is a DVR with prime \(p\) and \(t\geq 0\), then \(\text{rep}(S,R,t)\) is the category of objects \(U=(U_0,U_i:i\in S)\) where \(U_0\) is a finitely generated free \(R\)-module, each \(U_i\) is a summand of \(U_0\), \(U_i\subseteq U_j\) whenever \(i\leq j\) in \(S\) and \(p^tU_0\subseteq\sum\{U_i:i\in S\}\). Most of this chapter describes work by the author and M. Dugas characterizing representation type of such categories.

Chapter 5 treats almost completely decomposable (acd) groups, a class that has been the subject of considerable study in recent years. Here, the approach takes advantage of the results on Butler groups in Chapter 3 and the representations over DVR’s discussed in Chapter 4. Frequent use is made of functors that take acd groups to representations over \(Z_p\). Near isomorphism is the proper context in which to study acd groups, and the standard tools are developed and tied to representation concepts. In particular, the author provides a streamlined proof of his well-known result that indecomposability is preserved under near isomorphism (Corollary 5.1.8).

In Chapter 6, we return to representations of posets over fields to study projectives, injectives, Coxeter correspondences and almost split sequences. The application of the Coxeter functors to projectives and injectives gives rise to preprojectives and preinjectives. These in turn can be used, for example, to describe all the indecomposable representations, over an arbitrary field, of the four-element antichain. The results are translated to Butler groups in Chapter 7, including a discussion of balanced projectives, balanced injectives, endomorphism rings and the so-called “bracket groups”. A bracket group is the torsion-free quotient of a finite rank completely decomposable group by a rank-one group. Surprising even in the context of Butler groups, the bracket groups can be classified up to isomorphism.

The book concludes in Chapter 8 with some applications to torsion-free modules over DVR’s and finite valuated groups.

This book provides a rich resource for a variety of readers. As noted above, it offers an excellent and much-needed introduction to the subject of representations of posets. It could readily be used for a second year graduate course in algebra, with many tools and suggestions for cutting edge research. Finally, it is a handy reference for anyone interested in the study of torsion-free Abelian groups of finite rank or related questions in 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 of finite rank (henceforth abbreviated tffr groups). The connection between tffr groups and representations is established in Chapter 3, which contains a discussion of Butler groups and a category equivalence between categories of Butler groups and \(Q\)-representations of associated posets. Chapter 2, in the meantime, discusses fundamental tools such as quasi-isomorphism, isomorphism at \(p\) and near isomorphism. There is also a section on stable range conditions and cancellation properties (based on Arnold’s 1981 book), and one on endomorphism rings of tffr groups, including Corner’s Theorem and the Beaumont-Pierce Principal Theorem.

A Butler group is a homomorphic image of a finite direct sum of rank-1 groups, namely subgroups of \(Q\). Butler groups have proved remarkably tractable, in strong contrast to tffr groups in general. They were originally defined and studied by Köhler and Butler in the mid-60’s. Butler, for example, showed that a tffr group is the homomorphic image of a finite direct sum of rank-1 groups if and only if it is a pure subgroup of a finite direct sum of rank ones. Dave Arnold was largely responsible for “popularizing” Butler groups through several papers and his 1981 book. If \(G\) is a Butler group then \(G\) has a finite typeset that is contained in a finite lattice of types \(L\). The category of Butler groups with typeset contained in \(L\) and morphism sets given by \(Q\otimes\text{Hom}(G,H)\), is equivalent to the representations over \(Q\) of the poset \(S\) defined by taking the join-irreducible elements of \(L\) with the opposite ordering imposed. This seminal result allows one to prove theorems about Butler groups and quasi-homomorphisms while working on the familiar ground of vector spaces over \(Q\). Since most results don’t depend on the field, one ends up with much more general theorems in the representation setting. This fact seems to have been under-utilized by the Abelian group community. After a summary of basic results, Chapter 3 concludes with some material on countable Butler groups, including the important characterization that \(G\) is Butler if and only if \(\text{Bext}(G,T)=0\) for all torsion groups \(T\).

In Chapter 4, we switch back to representations, this time over a discrete valuation ring (DVR), with \(Z_p\) being the model. If \(S\) is a poset, \(R\) is a DVR with prime \(p\) and \(t\geq 0\), then \(\text{rep}(S,R,t)\) is the category of objects \(U=(U_0,U_i:i\in S)\) where \(U_0\) is a finitely generated free \(R\)-module, each \(U_i\) is a summand of \(U_0\), \(U_i\subseteq U_j\) whenever \(i\leq j\) in \(S\) and \(p^tU_0\subseteq\sum\{U_i:i\in S\}\). Most of this chapter describes work by the author and M. Dugas characterizing representation type of such categories.

Chapter 5 treats almost completely decomposable (acd) groups, a class that has been the subject of considerable study in recent years. Here, the approach takes advantage of the results on Butler groups in Chapter 3 and the representations over DVR’s discussed in Chapter 4. Frequent use is made of functors that take acd groups to representations over \(Z_p\). Near isomorphism is the proper context in which to study acd groups, and the standard tools are developed and tied to representation concepts. In particular, the author provides a streamlined proof of his well-known result that indecomposability is preserved under near isomorphism (Corollary 5.1.8).

In Chapter 6, we return to representations of posets over fields to study projectives, injectives, Coxeter correspondences and almost split sequences. The application of the Coxeter functors to projectives and injectives gives rise to preprojectives and preinjectives. These in turn can be used, for example, to describe all the indecomposable representations, over an arbitrary field, of the four-element antichain. The results are translated to Butler groups in Chapter 7, including a discussion of balanced projectives, balanced injectives, endomorphism rings and the so-called “bracket groups”. A bracket group is the torsion-free quotient of a finite rank completely decomposable group by a rank-one group. Surprising even in the context of Butler groups, the bracket groups can be classified up to isomorphism.

The book concludes in Chapter 8 with some applications to torsion-free modules over DVR’s and finite valuated groups.

This book provides a rich resource for a variety of readers. As noted above, it offers an excellent and much-needed introduction to the subject of representations of posets. It could readily be used for a second year graduate course in algebra, with many tools and suggestions for cutting edge research. Finally, it is a handy reference for anyone interested in the study of torsion-free Abelian groups of finite rank or related questions in representations of posets.

Reviewer: C.Vinsonhaler (Storrs)

##### MSC:

16G20 | Representations of quivers and partially ordered sets |

16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |

20K15 | Torsion-free groups, finite rank |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20K40 | Homological and categorical methods for abelian groups |

20K25 | Direct sums, direct products, etc. for abelian groups |