Invariant differental forms on compact nilmanifolds

Let N=G/ be a compact nilmanifold, G a connected, simply connected, nilpotent Lie group with its discrete subgroup and Lie algebra . Let I* ( ) denote the invariant differential forms on .If I* ( ) H* ( ) is an injective map, then G is abelian and N is a torus. Furthermore, N has a formal minimal model. If N is an even-dimensional compact nilmanifold, it has a Kähler structure and invariant symplectic structure if and only if I* ( ) H* ( ) is injective.     

Tissus linéaires et théorèmes d'algébrisation de type Abdel-inverse et Reiss-inverse

A d-web in ( ,0) is given by d complex analytic foliations of codimension one in ( ,0) which are in general position. A d-web in ( ,0) is linear if all the leaves are (pieces of) hyperplanes in and is algebraic if it is associated, by duality, to a nondegenerate algebraic curve in of degree d. We characterize linear webs in ( ,0). We give explicit conditions under which a linear d-web in ( ,0) is algebraic and we obtain equations for in this case. Some related problems are discussed and some questions are posed.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Rings of regular functions on nilpotent orbits and their covers

Summary LetG be a complex semisimple algebraic group with Lie algebra . Let be a nilpotentG-orbit, its ring of regular functions. We derive a formula for as aG-module and prove some partial results on a cover of . We then relate this formula to various existing multiplicity formulas forK-types in Harish-Chandra bimodules ofG.Supported by National Science Foundation Grant DMS-8505550     

Functions of translation type and functorial properties of Segal algebras II

In this paper we investigate functorial properties of the Segal algebra which consists of all functionsf in Wiener's algebra onG with Fourier transform in Wiener's algebra on the dual group . Especially may serve as a very large and natural domain for Poisson's formula. Moreover, there is introduced a Segal algebraE ₀(G) containing as a subspace, but still eachfE ₀(S) satisfies Poisson's formula.     

On the isomorphisms and automorphism groups of circulants

Denote byC _n(S) the circulant graph (or digraph). LetM be a minimal generating element subset ofZ _n, the cyclic group of integers modulon, and In this paper, we discuss the problems about the automorphism group and isomorphisms ofC _n(S). When M S , we determine the automorphism group ofC _n(S) and prove that for any T if and only ifT = S, where is an integer relatively prime ton. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Bounds for the lattice point enumerator

For the lattice point enumerator of a lattice and a convex body K we give bounds in terms of the intrinsic volumes of K and of minimal determinants of . The intrinsic volumes are the normalized Minkowski quermassintegrals and the minimal determinants are analogous functionals of .     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

A characterization of 3-local geometry of M(24)

The largest Fischer 3-transposition group M(24) acts flag-transitively on a 3-local incidence geometry (M(24)) which is a c-extension of the dual polar space associated with the group O ₇(3). The action of the simple commutator subgroup M(24) is still flag-transitive. We show that (M(24)) is characterized by its diagram under the flag-transitivity assumption. The result implies in particular that (M(24)) is simply connected. The geometry (M(24)) appears as a subgeometry in the Buekenhout-Fischer 3-local geometry (F ₁) of the Monster group. The simple connectedness of (M(24)) has played a crucial role in the characterization of (F ₁), which has been achieved recently. When determining the possible structure of the parabolic subgroups we have used an unpublished pushing-up result by U. Meierfrankenfeld.Dedicated to Professor B. Fischer on the occasion of his sixtieth birthday     

Symmetric planes with non-classical tangent translation planes

We construct symmetric planes associated with an arbitrary locally compact connected nearfield . If is a proper nearfield, i.e. {;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet     

Sechzehndimensionale lokalkompakte Translationsebenen,deren Kollineationsgruppe G2 enthält

New results of Salzmann and Hubig say that a 16-dimensional (locally) compact topological projective plane in which the group of continuous collineations has dimension 40 is a translation plane. It is therefore important to determine all 16-dimensional locally compact translation planes with dim 40. From previous work of the author ([10]), it is known that such a plane is either the classical octonion plane, or dim =40 and contains a subgroup isomorphic to the compact exceptional group G₂, but no larger compact simple subgroup. In the present paper, all planes satisfying the latter property more generally with dim 38 are explicitly determined. Together with the classification of all 16-dimensional locally compact translation planes in which contains Spin(7) given by the author in [8], one thus knows all 16-dimensional locally compact translation planes with containing G₂ and dim 38. Via suitable Baer subplanes, the classification makes use of analogous results for 8-dimensional planes ([7]).

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Meinem verehrten Lehrer Helmut Salzmann zum 60. Geburtstag     

Operators of fibered groups

Let (G, ) denote a finite groupG with fibration . The group Aut (G, ) of operators of (G, ) is closely related to the group of collineations of (G, ). In this paper we investigate the geometric properties imposed on (G, ) by requiring that Aut (G, )=AutG. We find that in many instances this algebraic property restricts the geometry to a very special form.Dedicated to Professor Helmut Karzel on the occasion of his 60th birthday.     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.     

Sub-weakly Embedded Singular and Degenerate Polar Spaces

We show that every sub-weak embedding of any singular (degenerate or not) orthogonal or unitary polar space of non-singular rank at least 3 in a projective space PG , a commutative field, is the projection of a full embedding in some subspace PG of PG , where PG contains PG and is a subfield of . The same result is proved in the symplectic case under the assumption that the field over which the polarity is defined is perfect if the characteristic is 2 and if each secant line of the embedded polar space contains exactly two points of . This completes the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces (singular or not; degenerate or not) of non-singular rank at least 3 and defined over a commutative field , where in the characteristic 2 case is perfect if the polar space is symplectic and the degree of the embedding is 2.     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

On nets of orderq 2 and degreeq+1 admitting GL(2,q)

In this paper we consider finite nets of orderq ² and degreeq + 1 which admit GL(2,q). Our main result says that if a net of orderq ² and degreeq + 1 admits a collineation group with a point-regular normal subgroupT such that /T GL(2,q), then is isomorphic to a regulus net, a twisted regulus net, a Hering net, or . Except in the last one, each of them corresponds to a surface in PG(3,q) obtained from a homogeneous polynomial in two variables.     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

Reliability bounds in DFRA class with known mean and variance

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