08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Dominant dimensions of finite dimensional algebras(2012) Abrar, Muhammad; König, Steffen (Prof. Dr. rer. nat.)We study the dominant dimensions of three classes of finite dimensional algebras, namely hereditary algebras, quotient algebras of trees and serial algebras. We see that a branching vertex plays a key role to establish that the dominant dimension (dom.dim) of hereditary algebras (quivers) is at most one. We define arms of a tree and split trees into two classes: trees without arms and trees with arms. Like hereditary algebras, it turns out that the dominant dimension of the quotient algebras of trees can not exceed one. For serial algebras A associated to linearly oriented quiver with n vertices, we give lower and upper bounds of dom.dimA, and show that the bounds are optimal. It is also shown that some of the algebras A satisfy the conditions in the higher dimensional version of the Auslander's correspondence. Further we consider serial algebras corresponding to one-oriented-cycle quiver Q with n vertices, and give optimal bounds for a special subclass of these algebras. We conjecture that for any non self-injective quotient algebra A of Q dom.dimA is at most 2n-3, where the number of vertices n is bigger than 2.. Finally, we construct few examples of algebras having large (finite) dominant dimensions.Item Open Access From simplicial groups to crossed squares(2022) Asiki, Natalia-MariaSimplicial groups are defined to be contravariant functors from the simplex category to the category of groups. The truncation functor maps a simplicial group to a [2,0]-simplicial group, satisfying the Conduché condition. A functor from the category of [2,0]-simplicial groups to the category of crossed squares is constructed, following Porter. It is shown that the latter functor is not an equivalence of categories. In addition, Loday's variant of the resulting crossed square is constructed and shown to be isomorphic to Porter's variant.Item Open Access Representations of Hecke algebras of Weyl groups of type A and B(2001) Lipp, Johannes; Dipper, Richard (Prof. Dr.)The knowledge of the decomposition numbers of Hecke algebras associated to Weyl groups is very useful in the representation theory of finite groups of Lie type since the decomposition matrix of such an algebra embeds into that of the corresponding group. In the investigation of the Hecke algebras themselves, generic constructions - that is, constructions independent of the coefficient ring and the parameters - are a helpful tool. This thesis contributes to those two aspects of the theory of Hecke algebras. The first part of this thesis is concerned with decomposition numbers of blocks of Hecke algebras of type A. In particular, we consider blocks having core (0) and weight 3. First, we derive an upper bound for the decomposition numbers of an arbitrary block. This is used to show that all the decomposition numbers of a block having core (0) and weight 3 are 0 or 1. That result in turn enables us to describe a combinatorial algorithm for their calculation. Furthermore, we show that the decomposition numbers of a block having core (0) and weight 3 depend only on the ordinary and the quantized characteristic of the coefficient field. Moreover, if the ordinary characteristic is neither 2 nor 3 then they are already determined by the quantized characteristic alone. In the second part of this thesis, we construct generic Specht series for Hecke algebras of type A and generic bi-Specht series for Hecke algebras of type B. These are series of right ideals in those algebras such that all subquotients are Specht modules respectively bi-Specht modules. The construction of the Specht series generalizes ideas from Dipper and James for symmetric groups and Hecke algebras of type A. In particular, generic bases for the so-called PK-modules are introduced. The derivation of the bi-Specht series makes use of the Specht series and general methods from Dipper and James for the investigation of Hecke algebras of type B.Item Open Access On unipotent Specht modules of finite general linear groups(2004) Brandt, Marco; Dipper, Richard (Prof. Dr.)Many outstanding problems in representation theory can be solved with a proper understanding of the irreducible unipotent modules for the finite general linear group GL_n(q). For each partition lambda of n there is a Specht module S^lambda for GL_n(q), defined over a field F in terms of the intersection of the kernels of certain homomorphisms. If F is a field of characteristic zero, then S^lambda is irreducible and {S^lambda | lambda is a partition of n} is a complete set of pairwise non-isomorphic irreducible unipotent modules for GL_n(q). If the characteristic of F is coprime to q, then, in general, S^lambda has a unique top composition factor D^lambda and the D^lambda's are the irreducible unipotent modules for GL_n(q). For each Specht module S^lambda, a generating element e_lambda is known but, in general, no explicit basis for S^lambda as a vector space over F has been found. Richard Dipper and Gordon James made significant progress towards the construction of a basis of S^lambda for a two part partition lambda. This thesis is based on their work and further develops and improves the techniques introduced by them. We consider bases of S^lambda which satisfy additional properties. Namely, we define the concept of a standard basis of S^lambda and the concept of a basis with corresponding polynomials of S^lambda. Against the background of these definitions we examine the Specht modules S^lambda for lambda=(n-m,m) and lambda=(2,2,2). In the case of the two part partition lambda=(n-m,m) we find a standard basis of S^lambda if 0 < m < 12 and we conjecture that the techniques used there will work in general for lambda=(n-m,m). For lambda=(2,2,2) we construct a basis with corresponding polynomials of S^lambda.Item Open Access Gendo-Frobenius algebras and comultiplication(2022) Yırtıcı, ÇiğdemGendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.Item Open Access Algebraic analogues of resolution of singularities, quasi-hereditary covers and Schur algebras(2021) Santos Cruz, Tiago Miguel dos; Koenig, Steffen (Prof. Dr.)In algebraic geometry, a resolution of singularities is, roughly speaking, a replacement of a local commutative Noetherian ring of infinite global dimension by a local commutative Noetherian ring of finite global dimension. In representation theory, an analogous problem is asking to resolve algebras of infinite global dimension by algebras of finite global dimension. In addition, such resolutions should have nicer properties to help us study the representation theory of algebras of infinite global dimension. This motivates us to take split quasi-hereditary covers as these algebraic analogues of resolutions of singularities and measure their quality using generalisations of dominant dimension and deformation results based on change of rings techniques. The Schur algebra, SR(n,d) with n ≥ d, together with the Schur functor is a classical example of a split quasi-hereditary cover of the group algebra of the symmetric group, RSd , for every commutative Noetherian ring R. The block algebras of the classical category O, together with their projective-injective module, are split quasi-hereditary covers of subalgebras of coinvariant algebras. In this thesis, we study split quasi-hereditary covers, and their quality, of some cellular algebras over commutative Noetherian rings. The quality of a split quasi-hereditary cover can be measured by the fully faithfulness of the Schur functor on standard modules and on m-fold extensions of standard modules. Over fields, the dominant dimension controls the quality of the split quasi-hereditary cover of KSd formed by the Schur algebra SK(n,d) and the Schur functor. In particular, this quality improves by increasing the characteristic of the ground field. To understand the integral cases, the classical concept of dominant dimension is not useful since in most cases there are no projective-injective modules. Using relative homological algebra, we develop and study a new concept of dominant dimension, which we call relative dominant dimension, for Noetherian algebras which are projective over the ground ring making this concept suitable for the integral setup. For simplicity, we call Noetherian algebras which are projective over the ground ring just projective Noetherian algebras. While developing the theory of relative dominant dimension, we generalize the Morita-Tachikawa correspondence for projective Noetherian algebras and we prove that computations of relative dominant dimension over projective Noetherian algebras can be reduced to computations of dominant dimension over finite-dimensional algebras over algebraically closed fields. Using relative dominant dimension, concepts like Morita algebras and gendo-symmetric algebras can be defined for Noetherian algebras. We compute the relative dominant dimension of Schur algebras SR(n,d) for every commutative Noetherian ring R. Using such computations together with deformation results that involve the spectrum of the ground ring R we determine the quality of the split quasi-hereditary covers of RSd , (SR(n,d), V⊗d) formed by the Schur algebra SR(n,d) and the Schur functor HomSR(n,d)(V⊗d,−): SR(n,d)-mod → RSd-mod for all regular Noetherian rings. Over local commutative regular rings R, the quality of (SR(n,d), V⊗d) depends only on the relative dominant dimension and on R containing a field or not. For this cover, the quality improves compared with the finite-dimensional case whenever the local commutative Noetherian ring does not contain a field. This theory is also applied to q-Schur algebras and Iwahori-Hecke algebras of the symmetric group. In full generality, we prove that the quality of a split quasi-hereditary cover of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple B-modules. Other split quasi-hereditary algebras that we study in this thesis are deformations of block algebras of the Bernstein-Gelfand-Gelfand category O of a semi-simple Lie algebra. These deformations provide split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras. We compute the relative dominant dimensions of these block algebras and we determine the quality of these covers. In these deformations, the quality dramatically improves compared with the finite-dimensional case. Using approximation theory to generalize once more the concept of dominant dimension to relative dominant dimension with respect to direct summands of the characteristic tilting module, we find new split quasi-hereditary covers. In particular, the relative dominant dimension of a characteristic tilting module of SR(n,d) with respect to V⊗d is a lower bound of the quality of a split quasi-hereditary cover of the cellular algebra EndSR(n,d)(V⊗d)op, independent of the natural numbers n and d. This split quasi-hereditary cover involves the Ringel dual of the Schur algebra SR(n,d). Using this technology for deformations of block algebras of the classical BGG category O of a semi-simple Lie algebra, we obtain a new proof for Ringel self-duality of the blocks of the classical BGG category O of a complex semi-simple Lie algebra. Here, the uniqueness of split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras with higher quality is the crucial factor to deduce Ringel self-duality.Item Open Access Dominant dimensions of algebras(2017) Marczinzik, René; König, Steffen (Prof. Dr.)Item Open Access l-groups and Bezout domains(2006) Yang, Yi Chuan; Rump, Wolfgang (Prof. Dr.)Wir untersuchen die Strukturtheorie abelscher l-Gruppen im Hinblick auf ihre Beziehungen zur Teilbarkeit in Bezout-Bereichen und zu MV-Algebren. Hauptmotivation für die Arbeit ist die Jaffard-Ohm-Korrespondenz zwischen abelschen l-Gruppen und Bezout-Bereichen, und D. Mundici's funktorielle Äquivalenz zwischen abelschen l-Gruppen mit starker Einheit und MV-Algebren. Mit Hilfe der Bewertungstheorie erhalten wir eine positive Antwort zum Problem von Conrad und Dauns (1969), ob ein Verbandsschiefkörper mit positiven Quadraten stets angeordnet ist. Als Gegenstück hierzu beweisen wir die Existenz gerichteter Algebren mit negativen Quadraten. Für eine beliebige l-Gruppe verallgemeinern wir die Charakterisierung von o-Idealen in Rieszschen Räumen auf l-Ideale (2004). Mittels der Mundici-Korrespondenz zwischen l-Gruppen und MV-Algebren erhalten wir eine negative Entscheidung einer Frage von Belluce (1986) über Prim-Annihilatoren in MV-Algebren. Für eine abelsche l-Gruppe G konstruieren wir eine dichte Einbettung von G in eine lateral vollständige abelsche l-Gruppe E(G) mittels Garbentheorie. Falls G archimedisch ist, beweisen wir, dass E(G) die laterale Vervollständigung von G ist, während dies im Allgemeinen falsch zu sein scheint. Als Nebenprodukt erhalten wir einen eleganten Beweis des Bernauschen Einbettungssatzes für Archimedische l-Gruppen. Ist G die Teilbarkeitsgruppe eines Bezout-Bereichs D, so zeigen wir, dass Spec(D) zum Spektrum Spec*(G) topologisch dual ist im Sinne von Hochster. Wir studieren die C-Topologie abelscher l-Gruppen mit Blick auf Bezout-Bereiche und MV-Algebren. Dabei korrigieren wir zwei Lemmata von Gusi\'c (1998), was zu einer Verallgemeinerung eines seiner Hauptergebnisse führt. Schliesslich beantworten wir eine Frage von Dumitrescu, Lequain, Mott und Zafrullah (2001), indem wir zeigen, dass die Jaffard-Ohm Korrespondenz für abelsche fast-l-Gruppen nicht gilt.Item Open Access Supercharacter theories for Sylow p-subgroups 3D4syl(q3), G2syl(q) and 2G2syl(32m+1)(2017) Sun, Yujiao; Dipper, Richard (Prof. Dr.)Item Open Access Desarguesian and geometric right ℓ-groups(2021) Dietzel, Carsten; König, Steffen (Prof. Dr.)