An Introduction to Quiver Representations

A quiver is finite directed graph, with loops and multiple edges allowed. Fixing a field \(K\) (which, for ease of exposition one may assume it is the field of complex numbers \(\mathbb C\)), a representation of a given quiver attaches a finite dimensional vector space to each vertex and a linear transformation to each arrow (edge) of the quiver.

The dimension vector of a representation of a quiver assigns to each vertex of \(Q\) the dimension of the corresponding vector space. A morphism between two representations of the same quiver is a collection of linear maps between vector spaces attached to the same vertex that make each corresponding square commute.

For each quiver \(Q\), the constructions given above define an abelian category \(\mathrm{Rep}(Q)\), with natural notions of zero object, subrepresentations, direct sums, kernels and cokernels. One initial result is the Krull-Remak-Schmidt theorem: every finite dimensional representation of a quiver is isomorphic to a direct sum of indecomposable representations of the same quiver. A direct consequence is that to classify all representations of a quiver it is enough to classify all its indecomposable representations.

Given a quiver \(Q\) its path algebra \(KQ\) is the \(K\)-vector space with a basis the set of all paths in the quiver and multiplication given by concatenation of paths. It turns out that the category of representations of \(K\)-algebras is equivalent to the category of representations of the quotient algebra \(KQ/I\), for some quiver \(Q\) and a two-sided ideal \(I\) of \(KQ\).

A few years after P. Gabriel started the development of the theory of quiver representations in the early 1970s, M. Auslander and I. Reiten introduced the methods of almost split sequences and irreducible morphims, which they used to construct certain quivers (the knitting algorithm) that are crucial to analyze the structure of algebras of finite or tame representation types.

For the classification of indecomposable representation algebras of wild type, methods of algebraic geometry, specifically from Geometric Invariant Theory, have provided a fertile approach: for every dimension vector \(\alpha\) of a representation \(V\) of a given quiver \(Q\) and each vertex \(x\) of \(Q\) one chooses an isomorphism \(V(x)={\mathbb C}^{\alpha(x)}\). Thus, for each arrow \(f\) of Q, with tail \(t(f)\) and head \(h(f)\), the linear transformation \(V(f)\) is an \(\alpha(h(f))\times \alpha(t(f))\) matrix with entries in the field \({\mathbb C}\). Then, the representation space of the quiver \(Q\) with dimension vector \(\alpha\) is the space \[\text{Rep}_{\alpha}(Q)=\prod_{\text{all arrows \(f\) of \(Q\)}}\text{Mat}_{\alpha(h(f)),\alpha(t(f)},\] where \(\text{Mat}_{\alpha(h(f)),\alpha(t(f)}\) is the space all of complex matrices of the given size. The natural action of the linear group \(\text{GL}_{\alpha(x)}\) on each \(\text{Mat}_{\alpha(h(f)),\alpha(t(f)}\), induces an action of the direct product \(\text{GL}_{\alpha}\) of the linear algebraic groups \(\text{GL}_{\alpha(x)}\) on the representation space \(\text{Rep}_{\alpha}(Q)\). There is a bijection between the isomorphism classes of representations of the quiver \(Q\) with dimension vector \(\alpha\) and the orbits of the action described above. As usually happens with orbit spaces, the closed orbits of this action don't pick all the representations of \(Q\), just the semisimple ones. Using Geometric Invariant Theory, A. D. King (Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. 2, Vol. 45, No. 180 (1994), pp. 515–530) proved that the GIT-quotient of the above action parametrizes only representations with certain stability conditions, for which it is indeed a coarse moduli space.

The exposition in the book under review could be divided in four parts. The first part, comprising chapters one to four, is mainly devoted to proper quiver representations, culminating with Gabriel’s theorem, truly a turning point in the history of representation theory.

The second part, chapters five to seven, focuses on the Auslander-Reiten theory of almost split sequences. Besides the foundations, here we find the construction of the Auslander-Reiten quiver and the functorial definition of the Coxeter functor. These constructions allow generalizations of Gabriel’s theorem to the tame case, and some of them are treated in these chapters. Moreover, a single chapter, eight, is devoted to formulation and proof of Kac’s theorem, which generalizes Gabriel’s theorem to arbitrary quivers.

The third part, chapters nine and ten, introduces the invariant theory methods that are required to deal with the wild case. As an application to representation theory, the authors prove King’s theorem characterizing stable and semistable representations using the Hilbert-Mumford criterion. Chapter 10, after reviewing some background on classical invariant theory, in particular the First Fundamental Theorem that describes generators for the space of invariants of the general linear group, the authors prove the Le Bruyn-Procesi theorem that gives a description of the invariant ring \({\mathbb C}[\text{Rep}_{\alpha}(Q)]^{\text{GL}_{\alpha}}\), for a quiver \(Q\) and a dimension vector \(\alpha\).

After reviewing some more background on the representation theory of the general linear group (weights, Young tableaux, Schur functors), the authors give the description of the ring of invariants of the normal subgroup of the general linear group \(\text{GL}_{\alpha}\) given by the direct product of the special linear groups \(\text{SL}_{\alpha(x)}\), for all vertices \(x\) of the quiver \(Q\), and acting on the representation algebra \(\text{Rep}_{\alpha}(Q)\). The semi-invariants involved in this description can be calculated using the Littlewood-Richardson rule. One of the main results in this chapter is that the ring of semi-invariants from quiver representations is spanned by certain invariants obtained by a method due to Schofield.

The fourth part, chapters eleven and twelve, focuses on certain combinatorial structures underlying quiver representations. Chapter eleven includes a study of exceptional quiver representations (Schur representations with trivial Ext), the canonical decomposition of a quiver representation, tilting modules and orthogonal categories. Chapter twelve, on cluster categories (the categorification of cluster algebras) gives an overview of recent work on representations of Artin algebras, of a combinatorial nature, with strong connections to seemingly remote areas of mathematics. The topics range from cluster combinatorics and decorated representations to the structure of the derived category of the abelian category of representations of a quiver and to cluster tilted algebras. There is quick overview of the required background on triangulated and derived categories, but a reader not already acquainted with them will find the going a little hard.

The exposition in the book is enriched with several well-chosen examples or detailed analysis of some cases that illustrate concepts or results. A few exercises at the end of the sections will certainly attract a motivated student. The book is accessible to a student with a solid background on linear algebra, including some homological algebra, and, for a few very specific instances in the text, a previous exposure to algebraic geometry.

The book under review joins a very healthy flow of monographs and textbooks on representation theory, whose recent additions have some overlapping with the one being reviewed. This new book sits between the elementary introduction by R. Schiffler Quiver Representations (Springer, 2014) and A. Kirillov’s more advanced monograph Quiver Representations and Quiver Varieties (AMS, 2016), but with emphasis on the connections between quiver representations and invariant theory.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.