On Auslander-Reiten components of blocks and self-injective biserial algebras

Erdmann, Karin; Skowroński, Andrzej

doi:10.1090/s0002-9947-1992-1144759-7

Cited by 64 publications

(24 citation statements)

References 29 publications

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“…We refer the reader to [3, (4.15.6)] for further details. For group algebras of finite groups, the possible tree classes and admissible groups were first determined by Webb [55], with refinements provided in [47,42,5,6,14].…”

Section: Auslander-reiten Components Of Simple G R -Modulesmentioning

confidence: 99%

Support varieties, AR-components, and good filtrations

Farnsteiner

Röhrle²

2009

Math. Z.

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Abstract. Let G be a reductive group, defined over the Galois field Fp with p being good for G. Using support varieties and covering techniques based on GrT -modules, we determine the position of simple modules and baby Verma modules within the stable Auslander-Reiten quiver Γs(Gr) of the r-th Frobenius kernel of G. In particular, we show that the almost split sequences terminating in these modules usually have an indecomposable middle term.Concerning support varieties, we introduce a reduction technique leading to isomorphismsfor baby Verma modules of certain highest weights λ, µ ∈ X(T ), which are related by the notion of depth. IntroductionIn the representation theory of finite-dimensional self-injective algebras, the stable AuslanderReiten quiver has proven to be an important homological invariant, which has been studied for group algebras of finite groups, reduced enveloping algebras of restricted Lie algebras and distribution algebras of infinitesimal group schemes. In the classical context of Frobenius kernels of reductive groups, the relevant algebras are usually of wild representation type, rendering a classification of their indecomposable modules a hopeless task. This fact notwithstanding, one does have a fairly good understanding of the connected components of the corresponding stable Auslander-Reiten quiver. It is therefore of interest to relate this information to certain classes of indecomposable modules, such as simple modules, Weyl modules or Verma modules, and to identify the position of the latter within the AR-quiver. The main problem in this undertaking is the lack of a suitable presentation of the underlying algebras, as required by the techniques of abstract representation theory. On the other hand, the theory of rank varieties and support varieties has seen considerable progress over the last years, so that one can hope to exploit these tools in the aforementioned context.In continuation of work begun in [22], we study in this article those Auslander-Reiten components of the algebras Dist(G r ) that contain simple modules or baby Verma modules. The underlying algebra Dist(G r ) consists of the distributions of the r-th Frobenius kernel of the smooth reductive group scheme G, defined over an algebraically closed field of positive characteristic p. The case r = 1, which pertains to restricted enveloping algebras of reductive Lie algebras, was settled in [22] by means of a detailed analysis of nilpotent orbits in rank varieties, leading to the consideration of groups of types SL(2) 1 ×SL(2) 1 , SL(3) 1 and SO(5) 1 . For r > 1, rank varieties are less tractable and the structure of the cohomology rings defining support varieties is more complicated. We address these problems by passage to G r T -modules, as described below. With regard to Auslander-Reiten theory, our main results can roughly be summarized as follows:Theorem. Let G be a smooth reductive group scheme, defined over F p . Given a character λ ∈ X(T ) and r ≥ 1, the following statements hold:

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Section: Auslander-reiten Components Of Simple G R -Modulesmentioning

confidence: 99%

Support varieties, AR-components, and good filtrations

Farnsteiner

Röhrle²

2009

Math. Z.

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“…Using results on special biserial algebras [12], we obtain the classification of the connected components of the stable Auslander-Reiten quiver of G for the case covered by the above result. In this context, the number r := ht(G/M(G)) turns out to be related to the rank of the exceptional tubes via = p r−1 .…”

Section: ) If B Possesses a Simple Module Of Dimension = P Then B Imentioning

confidence: 99%

Galois actions and blocks of tame infinitesimal group schemes

Farnsteiner

Skowroński

2007

Trans. Amer. Math. Soc.

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Abstract. Given an infinitesimal group G, that is defined over an algebraically closed field of characteristic p ≥ 3, we determine the block structure of the algebra of measures H(G) in case its principal block B 0 (G) is tame and the height of the factor group G/M(G) of G by its multiplicative center M(G) is at least two. Our results yield a complete description of the stable Auslander-Reiten quiver of H(G) along with a criterion for the domesticity of H(G).

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“…For general background concerning representation theory of algebras and selfinjective algebras applied here we refer to [1], [5], [6], [14] and [19].…”

Section: R Bocian and A Skowrońskimentioning

confidence: 99%

“…Then clearly A is a special biserial algebra whose bound quiver contains at most two primitive walks, and consequently A is of domestic type (see [6], [16]). Applying now [16] we infer that A is a selfinjective algebra of Euclidean type A m .…”

Section: R Bocian and A Skowrońskimentioning

confidence: 99%

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Symmetric special biserial algebras of euclidean type

Bocian

Skowroński

2003

Colloq. Math.

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Abstract. We classify (up to Morita equivalence) all symmetric special biserial algebras of Euclidean type, by algebras arising from Brauer graphs.Introduction and the main result. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finitedimensional K-algebra with identity, which we shall assume (without loss of generality) to be basic and connected. For an algebra A, we denote by mod A the category of finite-dimensional right A-modules and by D the standard duality Hom K (−, K) on mod A. The Cartan matrix C A of A is the matrix (dim K Hom A (P i , P j )) 1≤i,j≤n for a complete family P 1 , . . . , P n of pairwise nonisomorphic indecomposable projective A-modules.An algebra A is called selfinjective if A ∼ = D(A) in mod A, that is, the projective A-modules are injective. Further, A is called symmetric if A and D(A) are isomorphic as A-bimodules. For a selfinjective algebra A, we denote by Γ s A the stable Auslander-Reiten quiver of A, obtained from the Auslander-Reiten quiver Γ A of A by removing all projective modules and the arrows attached to them. We also note that if A is symmetric then the Auslander-Reiten translation τ A = D Tr in mod A is the square Ω 2 A of the Heller syzygy operator Ω A . An important class of selfinjective algebras is formed by the algebras of the form B/G, where B is the repetitive algebra [8] (locally finite-dimensional, without identity)

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On Auslander-Reiten components of blocks and self-injective biserial algebras

Cited by 64 publications

References 29 publications

Support varieties, AR-components, and good filtrations

Support varieties, AR-components, and good filtrations

Galois actions and blocks of tame infinitesimal group schemes

Symmetric special biserial algebras of euclidean type

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