08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Generalized polygons with doubly transitive ovoids(2013) Krinn, Boris; Stroppel, Markus (apl. Prof. Dr. rer. nat.)This thesis studies finite generalized quadrangles and hexagons which contain an ovoid. An ovoid is a set of mutually opposite points of maximum size. The first objective of the present work is to show that a generalized quadrangle or hexagon is a classical polygon if it contains an ovoid in the case of quadrangles or an ovoid-spread pairing in the case of hexagons, such that a group of given isomorphism type acts on the polygon in such a way that it leaves the ovoid invariant. The groups in use here are Suzuki and Ree groups. The second part of the thesis is devoted to the problem of determining all groups which can act on a generalized quadrangle or generalized hexagon in such a way that they operate doubly transitively on an ovoid of this polygon. It will turn out that these groups are essentially the known examples of groups acting on a classical or semi-classical ovoid of a classical polygon. In the first two chapters, meant as an introduction to the problem, constructions of the relevant generalized polygons are given and known results concerning the existence question of ovoids in known generalized quadrangles and hexagons are collected. In the following chapter, the known polarity of the symplectic quadrangle is presented and it is shown that the symplectic quadrangle can be reconstructed from the action of the Suzuki group on the absolute elements of this polarity. Then we show that any generalized quadrangle which contains an ovoid, such that a Suzuki group acts on the ovoid, is isomorphic to the symplectic quadrangle. In Chapter 4, analogous to the approach used for symplectic quadrangles, the known polarity of the split Cayley hexagon is described and it is shown that the split Cayley hexagon can be reconstructed from the action of the Ree group on the absolute points of the polarity. Then we show that any generalized hexagon that contains an ovoid-spread-pairing, on which a Ree group acts, is isomorphic to the classical split-Cayley hexagon. Both of these results are achieved without the use of classification results. The last three chapters are devoted to the problem of determining all groups which can act doubly transitively on an ovoid of a generalized quadrangle or generalized hexagon. This chapter uses the classification of finite simple groups (via the classification of finite doubly transitive groups). For hexagons the result is that only unitary groups and Ree groups are possible. This result, together with the one obtained in the previous chapter and a theorem by Joris De Kaey provides that the generalized hexagon is classical and the ovoid is classical if the ovoid belongs to an ovoid-spread-pairing. The result for quadrangles is less smooth. A further restriction on the order of the quadrangle is needed, namely that the number of points per line and the number of lines per point coincide and that this number is a prime power. This was not necessary in the case of the hexagons. With this additional assumption, we show that only orthogonal groups or Suzuki groups can act on these ovoids.Item Open Access Descriptions of some double Burnside rings(2017) Krauß, NoraThe double Burnside R-algebra B_R(G,G) of a finite group G with coefficients in a commutative ring R has been introduced by S. Bouc. It is R-linearly generated by finite (G,G)-bisets, modulo a relation identifying disjoint union and sum. Its multiplication is induced by the tensor product. It contains the bifree double Burnside R-algebra B_R^Delta(G,G) generated by bifree finite (G,G)-bisets. Let S_n denote the symmetric group on n letters. For R in {Q, Z, Z_(2), F_2, Z_(3), F_3}, we calculate B_R(S_3,S_3) and B_R^Delta(S_4,S_4).Item Open Access Basic representation theory of crossed modules(2018) Truong, MonikaA group corresponds to a topological space with one nontrivial homotopy group. A crossed module corresponds to a topological space with two nontrivial homotopy groups. In classical group theory, Cayley's Theorem constructs for every group G an injective group morphism to the symmetric group S_G. For a crossed module V, we have a similar statement. For every category C, we have the symmetric crossed module S_C. For every crossed module V, we construct an injective crossed module morphism to the symmetric crossed module S_VCat. Suppose given an R-linear category M. On the one hand, we obtain the invertible monoidal category Aut_R(M) by means of category theory. On the other hand, we have the symmetric crossed module S_M as in the Cayley context. In S_M, we have the crossed submodule Aut^CM_R(M) containing only the R-linear elements of S_M. We consider the corresponding invertible monoidal category (Aut^CM_R(M))Cat. We show that there exists a monoidal isofunctor Real_M : (Aut^CM_R(M))Cat -~-> Aut_R(M). This means that starting with M, we obtain essentially the same object via crossed module theory as via category theory. A representation of a group G on an R-module N is given by a group morphism G -> Aut_R(N). Analogously, a representation of a crossed module V on an R-linear category M is given by a crossed module morphism V -> Aut^CM_R(M). We begin to study the representation theory of crossed modules.Item Open Access The discriminant embedding(2021) Döring, Svea RikeWe construct a map from the complex projective n-space into a (3(n+1)n/2-1)-dimensional sphere, called discriminant embedding. In case n = 1, the discriminant embedding is the Riemann sphere map. To show that the discriminant embedding is in fact an immersion, we calculate the determinant of a matrix resulting from its Jacobian. This is related to constructions of G. Mannoury and B. A. Fuks.Item Open Access A note on the Fröhlich dynamics in the strong coupling limit(2021) Mitrouskas, DavidWe revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state ϕ0 ⊗ξα, where ϕ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t = O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation.Item Open Access The minimal projective resolution of Z_(2) over Z_(2)S_4(2017) Nitsche, SebastianWe consider the trivial module Z_(2) over the group ring Z_(2)S_4. Using the Wedderburn image of Z_(2)S_4, a minimal projective resolution of Z_(2) over Z_(2)S_4 with regular behavior is constructed. We give a closed formula for the projective terms and the differentials. The minimal projective resolution is used to calculate the cohomology groups of S_4 over Z_(2). In 1974, Thomas gave a description of the cohomology ring of S_4 as a factor ring of a polynomial ring. As far as we were able to compare both using Magma, our calculation is in accordance with his result.Item Open Access Resolution and realisation functors(2020) Stein, Nico; Künzer, Matthias (Priv.-Doz. Dr.)Item Open Access A sieve formula for chains of p-subgroups : extending Wielandt’s proof of Sylow-Frobenius to a congruence modulo p^(ℓ+1)(2024) Schwesig, EliasGiven a finite group G and a prime p, we establish the sieve formula, which is a congruence containing as summands numbers of chains of p-subgroups of G of certain orders. This generalises the Theorem of Sylow-Frobenius. Its name stems from the sieve formula from set theory because of formal similarities.Item Open Access The resolution equivalence for n-complexes(2022) Klein, VeronikaAn injective resolution equivalence is constructed which generalizes the construction from classical homological algebra to n-complexes. The construction proceeds by showing that a functor in the converse direction is an equivalence. The injective resolution functor is defined to be its inverse.Item Open Access Universelle Konstruktionen für Relationen(2024) Bechtel, JonasIn der Arbeit wird die Kategorie der Mengen und Relationen betrachtet, deren Objekte Mengen und deren Morphismen Relationen sind. Gegeben seien zwei Relationen, welche als Ursprungsmenge die gleiche Menge aufweisen. Für diese Relationen können je nach Situation die Eigenschaften linkstotal, rechtstotal, linkseindeutig und rechtseindeutig gefordert sein. Für die sich hieraus ergebenden Ausgangssituationen wurde untersucht, wann stets ein Pushout konstruiert werden kann und für welche Situationen es Gegenbeispiele zur Existenz eines Pushouts gibt. Existiert ein Pushout, so ist in der Arbeit auch die Pushout-Konstruktion angegeben. Des Weiteren wurde die Faktorisierung einer Relation über ihren Graphen betrachtet und hierfür eine universelle Konstruktion angegeben.