Auslander-Reiten Quiver Research Articles

Auslander-Reiten Sequences for Gorenstein Rings of Dimension One

Let $R$ be a complete local Gorenstein ring of dimension one, with maximal ideal m. We show that if $M$ is a Cohen-Macaulay $R$-module which begins an AR-sequence, then this sequence is produced by a particular endomorphism of m corresponding to a minimal prime ideal. We apply this result to determining the shape of some components of Auslander-Reiten quivers.

Auslander–Reiten components of symmetric special biserial algebras

We provide a combinatorial algorithm for constructing the stable Auslander–Reiten component containing a given indecomposable module of a symmetric special biserial algebra using only information from its underlying Brauer graph. We also show that the structure of the Auslander–Reiten quiver is closely related to the distinct Green walks of the Brauer graph and detail the relationship between the precise shape of the stable Auslander–Reiten components for domestic Brauer graph algebras and their underlying graph. Furthermore, we show that the specific component containing a given simple or indecomposable projective module for any Brauer graph algebra is determined by the edge in the Brauer graph associated to the module.

SIMPLE MODULES IN THE AUSLANDER–REITEN QUIVER OF PRINCIPAL BLOCKS WITH ABELIAN DEFECT GROUPS

Given an odd prime $p$ , we investigate the position of simple modules in the stable Auslander–Reiten quiver of the principal block of a finite group with noncyclic abelian Sylow $p$ -subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$ , we prove that simple modules in the principal block all lie at the end of their components.

AR-Components of domestic finite group schemes: McKay-Quivers and Ramification

For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander-Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of $SL(2)$. Moreover, for a normal subgroup scheme $\mathcal{N}$ of a finite group scheme $\mathcal{G}$, we show that there is a connection between the ramification indices of the restriction morphism $\mathbb{P}(\mathcal{V}_{\mathcal{N}})\rightarrow\mathbb{P}(\mathcal{V}_{\mathcal{G}})$ between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander-Reiten quivers.

Auslander-Reiten Numbers Games

Let \(\mathfrak {g}\) be a simple complex Lie algebra of types A n , D n , E n , and Q a quiver obtained by orienting its Dynkin diagram. Let λ be a dominant weight, and E(λ) the corresponding simple highest weight representation. We show that the weight multiplicities of E(λ) may be recovered by playing a numbers game Λ Q (λ), generalizing the well known Mozes game, constructing the orbit of λ under the action of the Weyl group W. The game board is provided by the Auslander-Reiten quiver Γ Q of Q. The game moves are obtained by constructing Nakajima’s monomial crystal M(λ) directly out of Γ Q . As an application, we consider Kashiwara’s parameterizations of the canonical basis. Let w 0 be a reduced expression of the longest element w 0 of W, adapted to a quiver Q of type A n . We show that a set of inequalities defining the string (Kashiwara) cone with respect to w 0, may be obtained by playing subgames of the numbers games Λ Q (ω i ) associated to fundamental representations.

Representations of Regular Trees and Invariants of AR-Components for Generalized Kronecker Quivers

We investigate the generalized Kronecker algebra 𝒦 r = kΓ r with r ≥ 3 arrows. Given a regular component 𝒞 of the Auslander-Reiten quiver of 𝒦 r , we show that the quasi-rank rk(𝒞) ∈ ℤ≤1 can be described almost exactly as the distance 𝒲(𝒞) ∈ ℕ0 between two non-intersecting cones in 𝒞, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality $$-\mathcal{W}(\mathcal{C}) \leq \text{rk}(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.$$ Utilizing covering theory, we construct for each n ∈ ℕ0 a bijection φ n between the field k and {𝒞∣𝒞 regular component, 𝒲(𝒞) = n}. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.

The derived category of an algebra with radical square zero

ABSTRACTKoszul duality and covering theory are combined to realize the bounded derived category 𝒟 of an algebra with radical square zero as a certain orbit category of the bounded derived category of finitely presented representations of an associated infinite quiver. As a consequence, the possible shapes of the connected components of the Auslander–Reiten quiver of 𝒟 are described.

The Auslander–Reiten Components Seen as Quasi-Hereditary Categories

Quasi-hereditary algebras were introduced by E. Cline, B. Parshall and L. Scott in order to deal with highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups. These categories have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander, and used in his proof of the first Brauer–Thrall conjecture and later used systematically in his joint work with I. Reiten on stable equivalence, as well as many other applications. Recently, functor categories were used by Martinez-Villa and Solberg to study the Auslander–Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category. We can think of the Auslander–Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category $$\mathrm {Mod}(\mathcal {C} )$$ , with $$\mathcal C$$ a component of the Auslander–Reiten quiver.

CLASSIFICATION OF INDECOMPOSABLE MODULES FOR FINITE GROUP SCHEMES OF DOMESTIC REPRESENTATION TYPE

We investigate the representation theory of domestic group schemes $$ \mathcal{G} $$ over an algebraically closed field of characteristic p > 2. We present results about filtrations of induced modules, actions on support varieties, Clifford theory for certain group schemes and applications of Clifford theory for strongly group graded algebras to the structure of Auslander-Reiten quivers. The combination of these results leads to the classification of modules belonging to the principal block of the group algebra k $$ \mathcal{G} $$ .

Composition of irreducible morphisms in quasi-tubes

We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander–Reiten quivers of artin algebras in relation with the powers of the radical of their module category.

Almost split sequences for polynomial GrT-modules and polynomial parts of Auslander–Reiten components

In [8], Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via GrT-modules to representations of the algebraic group G=GLn. We study analogues of these algebras and their Auslander–Reiten theory for reductive algebraic groups G and Borel subgroups B by considering the categories of polynomial representations of GrT and BrT as full subcategories of modGrT and modBrT, respectively. We show that every component Θ of the stable Auslander–Reiten quiver Γs(GrT) of modGrT whose constituents have complexity 1 contains only finitely many polynomial modules. For G=GL2,r=1 and T⊆G the torus of diagonal matrices, we identify the polynomial part of the stable Auslander–Reiten quiver of GrT and use this to determine the Auslander–Reiten quiver of the infinitesimal Schur algebras in this situation. For the Borel subgroup B of lower triangular matrices of GL2, the category of BrT-modules is related to representations of elementary abelian groups of rank r. In this case, we can extend our results about modules of complexity 1 to modules of higher Frobenius kernels arising as outer tensor products.

Representations of quivers over the algebra of dual numbers

The representations of a quiver Q over a field k (the kQ-modules, where kQ is the path algebra of Q over k) have been studied for a long time, and one knows quite well the structure of the module category modkQ. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when A=k[ϵ] is the algebra of dual numbers (the factor algebra of the polynomial ring k[T] in one variable T modulo the ideal generated by T2), thus to the Λ-modules, where Λ=kQ[ϵ]=kQ[T]/〈T2〉. The algebra Λ is a 1-Gorenstein algebra, thus the torsionless Λ-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen–Macaulay modules). They form a Frobenius category L, thus the corresponding stable category L_ is a triangulated category. As we will see, the category L is the category of perfect differential kQ-modules and L_ is the corresponding homotopy category. The category L_ is triangle equivalent to the orbit category of the derived category Db(modkQ) modulo the shift and the homology functor H:modΛ→modkQ yields a bijection between the indecomposables in L_ and those in modkQ. Our main interest lies in the inverse, it is given by the minimal L-approximation. Also, we will determine the kernel of the restriction of the functor H to L and describe the Auslander–Reiten quivers of L and L_.

ON COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS THAT CONTAIN HELLER LATTICES: THE CASE OF TRUNCATED POLYNOMIAL RINGS

Let $A$ be a truncated polynomial ring over a complete discrete valuation ring ${\mathcal{O}}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes {\mathcal{K}}$ is projective as an $A\otimes {\mathcal{K}}$-module, where ${\mathcal{K}}$ is the fraction field of ${\mathcal{O}}$. Then, we may define the stable Auslander–Reiten quiver of the category. We determine the shape of the components of the stable Auslander–Reiten quiver that contain Heller lattices.

Degrees of irreducible morphisms in generalized standard coherent almost cyclic components

ABSTRACTWe study the degrees of irreducible morphisms between indecomposable modules lying in generalized standard coherent almost cyclic components of Auslander-Reiten quivers of artin algebras.

On Sectional Paths in a Category of Complexes of Fixed Size

Fil: Chaio, Claudia Alicia. Universidad Nacional de Mar del Plata; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Mar del Plata; Argentina

On syzygies over 2-Calabi–Yau tilted algebras

We characterize the syzygies and co-syzygies over 2-Calabi–Yau tilted algebras in terms of the Auslander–Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen–Macaulay modules, the representation dimension of algebras and the Igusa–Todorov functions. In particular, we prove that the Igusa–Todorov dimensions of d-Gorenstein algebras are equal to d.For cluster-tilted algebras of Dynkin type D, we give a geometric description of the stable Cohen–Macaulay category in terms of tagged arcs in the punctured disc. We also describe the action of the syzygy functor in a geometric way. This description allows us to compute the Auslander–Reiten quiver of the stable Cohen–Macaulay category using tagged arcs and geometric moves.

On duality preservability of Auslander-Reiten quivers of derived categories and cluster categorie

We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. Meantime, we characterize the condition of generalized path algebras as cleft extensions of path algebras.

On the Postprojective Partitions and Components of the Auslander-Reiten Quivers

In this article, we shall investigate further the connections between the postprojective partition of an algebra and its Auslander–Reiten quiver.

Auslander–Reiten quiver of type D and generalized quantum affine Schur–Weyl duality

We first provide an explicit combinatorial description of the Auslander–Reiten quiver ΓQ of finite type D. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra Uq′(Dn+1(i))(i=1,2) and the quiver Hecke algebra RDn+1 associated to Dn+1(n≥3), by using the combinatorial description and the generalized quantum affine Schur–Weyl duality functor. As applications, we can prove that Dorey's rule holds for the category Rep(RDn+1) and prove an interesting difference between multiplicity free positive roots and multiplicity non-free positive roots.

Irreducible Morphisms and Locally Finite Dimensional Representations

Let \(\mathcal {A}\) be a Hom-finite additive Krull-Schmidt k-category where k is an algebraically closed field. Let \(\text {mod}\mathcal {A}\) denote the category of locally finite dimensional \(\mathcal {A}\)-modules, that is, the category of covariant functors \(\mathcal {A} \to \text {mod}k\). We prove that an irreducible monomorphism in \(\text {mod}\mathcal {A}\) has a finitely generated cokernel, and that an irreducible epimorphism in \(\text {mod}\mathcal {A}\) has a finitely co-generated kernel. Using this, we get that an almost split sequence in \(\text {mod}\mathcal {A}\) has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results to the study of rep(Q), the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of rep(Q).