232 research outputs found
On extremal quasi-modular forms after Kaneko and Koike: With an appendix of G. Nebe
No full textKaneko and Koike introduced the notion of extremal quasi-modular form and proposed conjectures on their arithmetic properties. The aim of this note is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The note ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalised extremal quasimodular form of weight 14 and depth 1
Orthogonal representations of finite groups
No full textThe subject of this thesis are orthogonal representations of finite groups. By this we mean a pair (V, q) where V is a KG-module and q is a G-invariant non-degenerate quadratic form on V . We restrict our considerations to finite groups G and totally real number fields K.The G-invariant quadratic forms on V form a K-vector space and we are mainly interested in the case where that vector space is one-dimensional. If it is, we call the KG-module V uniform.It is a well known result of ordinary representation theory that the isomorphism type of V is determined by its character χ. In other words, if V is uniform, χ determines a non-degenerate G-invariant quadratic form q on V up to scalar multiples, which begs the question: How can we determine the isometry class of q given the character χ?First results on this question were achieved by Gabriele Nebe, who devised a purely character-theoretic method to answer the above question in her habilitation thesis. However, Nebe’s method is only applicable under some favourable (and restrictive) conditions, so in general the problem is still open.This thesis contributes to this open question in several ways. First, the theoretical background and the known results are presented. After that we discuss a version of Clifford theory for orthogonal representations of normal subgroups and classify the orthogonal representations of semidirect products of the form C_p : C_(p−1). These results are then used for one of the main results of the thesis, namely the classification of the orthogonalrepresentations of the infinite series of groups SL_2(q) for prime powers q. We also discuss an independent topic, so-called Clifford orders, which are subrings of the classical Clifford algebra of a quadratic space. We define those subrings and describe some of their arithmetic properties.The thesis is concluded with a section of computational results for orthogonal representations of the finite simple groups contained in the "Atlas of finite groups"
Dense lattices as Hermitian tensor products
No full textUsing the Hermitian tensor product description of the extremal even unimodular lattice of dimension 72 found by Nebe in 2010 we show its extremality with the methods from Coulangeons article in Acta Arith. 2000
Dense lattices as Hermitian tensor products
No full textUsing the Hermitian tensor product description of the extremal even unimodular lattice of dimension 72 found by Nebe in 2010 we show its extremality with the methods from Coulangeons article in Acta Arith. 2000
Dual strongly perfect lattices
No full textThe search for the densest sphere packings in every dimension is a classical problem in mathematics. In dimension 3 this problem is know as the Kepler conjecture, and has only been proofed by Thomas Hales in 1998. If we additionally assume, that the centres of the spheres in the packing form a group, a so called lattice, the problem becomes much easier. Already in 1908 Voronoi developed an algorithm, which computes all local densest lattices. But the complexity of this algorithm makes an application in dimensions greater than 8 impossible. Thus the densest sphere packings associated to a lattice are known up to dimension 8 and in dimension 24, only because of the existence of the extremely dense Leech lattice. Venkov united special local densest lattices with the combinatorial concept of spherical designs, which was developed by Delsarte, Goethals and Seidel in 1977. A lattice is called strongly perfect, if its shortest vectors form a spherical 5-design. As Venkov proves, strongly perfect lattices assume local maxima of the density function. Examples for strongly perfect lattices are the Leech lattice, which was mentioned above, the Barnes-Wall lattice and the densest lattices in dimension 2, 4, 6, 7, 8. G. Nebe and B. Venkov completely classified the strongly perfect lattices up to dimension 12. Almost all strongly perfect lattices are dual strongly perfect, which means that their dual lattice is also strongly perfect. In dimension 14 Nebe and Venkov proved that there is exactly one dual strongly perfect lattice. This thesis continues the classification of dual strongly perfect lattices and shows, that in dimension 13 and 15 there are no dual strongly perfect lattices. The methods developed in this thesis allow to give a shorter proof for the classification in dimension 14, but are not sufficient to give a classification in dimension 16. In dimension 17 we impose the additional constraint for the classification and prove that there is no universally perfect lattice, which means that all non empty layers of L are spherical 5-designs. This property can be characterised with theta-series: all theta-series of L with harmonical coefficients up to degree 5 vanish. With the theta-transformation formula we get that the dual lattice of an universally perfect lattice is universally perfect, too. Therefore universally perfect lattices are in particular dual strongly perfect
ON EXTREMAL QUASI-MODULAR FORMS AFTER KANEKO AND KOIKE
No full textKaneko and Koike introduced the notion of extremal quasi-modular forms and proposed conjectures on their arithmetic properties. The aim of this paper is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The paper ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalized extremal quasi-modular form of weight 14 and depth one
Group rings over the p-adic integers
No full textThe thesis "Group Rings over the p-Adic Integers" is concerned with the representation theory of finite groups over the ring of p-adic integers (or, usually, an algebraic extension thereof). It deals with blocks of special linear groups of degree two, blocks of dihedral defect and blocks of symmetric groups. In each case the aim is to describe basic orders of such blocks, that is, orders of minimal rank in the Morita equivalence class of the order in question. This thesis combines arithmetic methods from the theory of orders in semi simple algebras with methods from modular representation theory and representation theory of algebras (in particular derived equivalences). The thesis is structured as follows: The first chapter deals with the necessary representation theoretic and order theoretic prerequisites. In the second chapter, a new method is introduced which allows a reduction of the problem of lifting an algebra over a field of characteristic p to an order over an appropriate p-adic ring to the analogous problem for an algebra which is derived equivalent to the original one. In the third chapter the aforementioned method is applied to blocks of dihedral defect. There is a classification of such blocks over an algebraically closed field of characteristic two. Some of the algebras occurring in this classification can be lifted to an essentially unique order by elementary means. By making use of derived equivalences between different algebras in the classification, a description for all blocks of dihedral defect with more than one simple module is obtained. This solves an open problem concerning certain unknown scalars occurring in the classification of blocks of dihedral defect over an algebraically closed field of characteristic two. The third chapter then applies the method developed in the second chapter to blocks of special linear groups of degree two in defining characteristic. It is shown that such blocks have an essentially unique lift to an order over an appropriate p-adic ring, provided the field of coefficients for the group ring is sufficiently large. The crucial fact that is exploited here is that the blocks under consideration are derived equivalent to blocks of group rings of upper triangular unipotent matrices. As a corollary, an open conjecture of Nebe concerning the basic orders of blocks of special linear groups of degree two is proved. The last chapter of the thesis deals with blocks of symmetric groups. It contains a modified version of Scopes' reduction and a description of basic orders of defect two blocks of symmetric groups
-conjugate weight enumerators and invariant theory
No full textLet be a field, a finite group of field automorphisms of ,
the -fixed field in and GL a finite matrix
group. Then the action of defines a grading on the symmetric algebra
of the -space which we use to introduce the notion of homogeneous
-conjugate invariants of . We apply this new grading in invariant
theory to broaden the connection between codes and invariant theory by
introducing -conjugate complete weight enumerators of codes. The main
result of this paper applies the theory from Nebe, Rains, Sloane to show that
under certain extra conditions these new weight enumerators generate the ring
of -conjugate invariants of the associated Clifford-Weil groups. As an
immediate consequence we obtain a result by Bannai etal that the complex
conjugate weight enumerators generate the ring of complex conjugate invariants
of the complex Clifford group. Also the Schur-Weyl duality conjectured and
partly shown by Gross etal can be derived from our main result
The limits of autobiography in Nebe nemá dno
No full textThis thesis, The limits of autobiography in Nebe nemá dno, explores the limits of the genre of autobiography, using the novel by Hana Androniková as an example. Using selected examples, it attempts to capture the motifs and references in which this novel transcends the defined conception of the autobiographical genre and thus stands on its borderline. Theoretical concepts dealing with the limits of the autobiographical genre and its contradictions is applied to this text. In its next part, this thesis follows selected story lines and contrasts them with real events in the author's life. The records and testimonies of the author herself, as well as those of her family and close friends, are also important material for this research.Tato práce Hranice autobiografie na příkladu románu Nebe nemá dno se zabývá hranicemi žánru autobiografie na příkladu románu Hany Andronikové. Na zvolených ukázkách se snaží postihnout motivy a odkazy, v nichž tento román překračuje definované pojímání autobiografického žánru a stojí tak na jeho pomezí. Na tento text jsou aplikovány teoretické koncepty zabývající se hranicemi autobiografického žánru a jeho rozporuplností. Tato práce ve své další části sleduje vybrané příběhové linie a staví je do kontrastu s reálnými událostmi v autorčině životě. Důležitým materiálem k tomuto výzkumu jsou i záznamy a výpovědi autorky samotné i její rodiny a blízkých přátel.Programme Electronic Culture and SemioticsProgram Elektronická kultura a semiotikaFaculty of HumanitiesFakulta humanitních studi
Restrictions of characters and Weil characters in symplectic and unitary groups
No full textWe consider the representation theory of finite symplectic and unitary groups and examine restrictions of unipotent characters to maximal parabolic subgroups stabilizing an isotropic line with the goal of finding an upper bound for the multiplicities of these restrictions which does not depend on the field over which the group is defined. We work towards a reduction to the case of unipotent cuspidal characters
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