Representations of Dihedral Groups Over a Field of Characteristic 2

Bondarenko, V. S.

doi:10.1070/sm1975v025n01abeh002197

Cited by 22 publications

(19 citation statements)

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“…Indeed any simple group with dihedral Sylow subgroup is isomorphic to one of them. Thus the classification of their modular representations could, in principle, be deduced from Ringel's classification [7] (also obtained by Bondarenko [1]) of the modular representations of the dihedral group. This would involve a careful examination of the Green correspondence.…”

mentioning

confidence: 99%

The indecomposable modular representations of certain groups with dihedral sylow subgroup

Donovan

Freislich

1978

Math. Ann.

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This paper classifies the indecomposable modules over a class of quasi-Frobenius algebras over a field k. These algebras have a k-basis described by a Brauer graph.The product of two of these basis elements is either another basis element or zero. The Brauer trees that arise in the study of blocks with cyclic defect group are a special case.The modules may be described as diagrams of vector spaces subject to certain relationships of commutativity and to certain composites being zero. In this interpretation the commutativity of k need not be assumed. As the motivation of this paper is the representation theory of certain finite groups, the presentation is for commutative k only. However no restriction on the characteristic or algebraic closure of k is made.A module is said to be uniserial if it has a unique composition series. A module M is said to be special if it has two uniserial submodules U and V such that Urn V is the socle of M and is simple, M/U and M/V are uniserial and U + V is the unique maximal submodule of M. In particular, uniserial and (by special convention) simple modules are special. If k is algebraically closed, finite dimensional kalgebras whose indecomposable projective modules are special can have their representation theory determined by the methods of this paper.As implied above, the modular representation theory of a block with cyclic defect group is a special case of the work in this paper. More significantly, let k be commutative, have characteristic 2 and contain a splitting field for the group G =$4,$5, A 7 or PSL(2,q), where q is an odd prime power. The indecomposable projective kG-modules are special, and so the indecomposable modular representations of kG are classified by the main theorem of this paper. Section 2 implicitly describes the structure of the indecomposable projective modules in the principal block in each ease. The non-principal blocks cause little difficulty. The details need some modification when k does not contain a splitting field.The groups mentioned above have dihedral Sylow subgroups. Indeed any simple group with dihedral Sylow subgroup is isomorphic to one of them. Thus the classification of their modular representations could, in principle, be deduced from Ringel's classification [7] (also obtained by Bondarenko [1]) of the modular

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confidence: 99%

The indecomposable modular representations of certain groups with dihedral sylow subgroup

Donovan

Freislich

1978

Math. Ann.

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“…Theorem 0. (See [9,16,3,5,4,18]) Apart from the indecomposable projective left Λmodules of type (b) above, the indecomposable representations in Λ -mod are either string or band modules (to be described next). Conversely, all strings and bands are indecomposable.…”

Section: Background On Special Biserial Algebrasmentioning

confidence: 99%

“…That the homological behavior of special biserial algebras should be understood so late in the game, and in slow increments at that, is somewhat surprising, as they constitute one of the most thoroughly investigated classes of tame algebras, next to the hereditary algebras based on (extended) Dynkin graphs. They have, in fact, developed into a showcase for representation-theoretic techniques, due to the combined facts that • they occur widely in contexts of interest (such as the representation theory of the Lorentz group and among blocks of group algebras in characteristic 2; see [9], [3], [16], [5] and [6], for instance) and • the structure of their indecomposable finite dimensional representations is fully understood; it is governed by two simple templates, strings and bands (for use in our arguments, we define them below). The firm grip on the finite dimensional representations was, in turn, extensively used towards understanding Auslander-Reiten quivers and numerous other aspects of special biserial algebras, while the homological analysis lagged behind.…”

Section: Introduction and Conventionsmentioning

confidence: 99%

Representation-Tame Algebras Need not be Homologically Tame

Huisgen-Zimmermann

2016

Algebr Represent Theor

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We show that even within the class of special biserial algebras, one of the most thoroughly studied classes of representation-tame finite dimensional algebras, the (left) big finitistic dimension may be strictly larger than the little. In fact, we find that the discrepancies Fin dim − fin dim fail to be bounded as traces the special biserial algebras. More precisely: For every positive integer r, we exhibit a special biserial algebra with the property that fin dim = r + 1, while Fin dim = 2r + 1.

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“…Theorem 0. (See [17,29,6,12,8]) The finitely generated string and band modules are precisely the indecomposable objects of Λ -mod.…”

Section: Prerequisites and Conventionsmentioning

confidence: 99%

“…In particular, it was proved there that this algebra -along with a class of close relatives -has tame representation type; in fact, its finite dimensional indecomposable representations were explicitly pinned down. In a sequence of articles by Ringel [29], Bondarenko [6], Donovan-Freislich [12], Butler-Ringel [8], and others, the class of algebras amenable to techniques derived from the Gelfand-Ponomarev archetype was subsequently found to be much larger and, moreover, to be related to further classical scenarios, such as the representation theory of dihedral groups. This development ultimately led to a well-rounded representationtheoretic picture of the extended class of algebras on which we concentrate here, the class of string algebras.…”

Section: Introductionmentioning

confidence: 99%

The homology of string algebras I

Huisgen-Zimmermann

Smalø

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

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Dedikert til vår venn og kollega Idun Reiten i andledning hennes sekstiårsdag Abstract. We show that string algebras are 'homologically tame' in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra Λ are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of Λ can be computed from a finite set of cyclic left ideals contained in the Jacobson radical. Second, our main result shows that the functorial finiteness status of the full subcategory P <∞ (Λ -mod) consisting of the finitely generated left Λ-modules of finite projective dimension is completely determined by a finite number of, possibly infinite dimensional, string modules -one for each simple Λ-module -which are algorithmically constructible from quiver and relations of Λ. Namely, P <∞ (Λ -mod) is contravariantly finite in Λ -mod precisely when all of these string modules are finite dimensional, in which case they coincide with the minimal P <∞ (Λ -mod)-approximations of the corresponding simple modules. Yet, even when P <∞ (Λ -mod) fails to be contravariantly finite, these 'characteristic' string modules encode, in an accessible format, all desirable homological information about Λ -mod.

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Representations of Dihedral Groups Over a Field of Characteristic 2

Cited by 22 publications

References 2 publications

The indecomposable modular representations of certain groups with dihedral sylow subgroup

The indecomposable modular representations of certain groups with dihedral sylow subgroup

Representation-Tame Algebras Need not be Homologically Tame

The homology of string algebras I

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