A module-theoretic approach to Clifford theory for blocks

This work concerns a generalization of Clifford theory to blocks of group-graded algebras. A module-theoretic approach is taken to prove a one-to-one correspondence between the blocks of a fully group-graded algebra covering a given block of its identity component, and conjugacy classes of blocks of a twisted group algebra. In particular, this applies to blocks of a finite group covering blocks of a normal subgroup.

     

Relative flux homomorphism in symplectic geometry

In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We also show that some quotients of the universal covering of the group of symplectomorphisms are stable under symplectic reduction.

     

On the homotopy invariance of torsion for covering spaces

We prove the homotopy invariance of torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the cohomology of the covering space vanishes, the homotopy invariance was established by Lück. We also give some applications of our results.

     

On Siegel modular forms of half-integral weights and Jacobi forms

We will establish a bijective correspondence between the space of the cuspidal Jacobi forms and the space of the half-integral weight Siegel cusp forms which is compatible with the action of the Hecke operators. This correspondence is based on a bijective correspondence between the irreducible unitary representations of a two-fold covering group of a symplectic group and a Jacobi group (that is, a semidirect product of a symplectic group and a Heisenberg group). The classical results due to Eichler-Zagier and Ibukiyama will be reconsidered from our representation theoretic point of view.

     

Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces

We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any sufficiently large unramified covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings.

     

On pro-unipotent groups satisfying the Golod-Shafarevich condition

We prove that a pro-unipotent group satisfying the Golod-Shafarevich condition contains a free non-abelian pro-unipotent group. Together with the result of A. Magid this implies that such a group is not linear.

     

A smooth invariant not a smooth function of the invariant polynomials

An explicit example is given of a smooth function invariant under a linear group action that is not a smooth function of the invariant polynomials.

     

Constructing Covering Codes with Given Automorphisms

Let K(n,r) denote the minimum cardinality of a binary covering code of length n and covering radius r. Constructions of covering codes give upper bounds on K(n,r). It is here shown how computer searches for covering codes can be sped up by requiring that the code has a given (not necessarily full) automorphism group. Tabu search is used to find orbits of words that lead to a covering. In particular, a code D found which proves K(13,1) 704, a new record. A direct construction of D given, and its full automorphism group is shown to be the general linear group GL(3,3). It is proved that D is a perfect dominating set (each word not in D is covered by exactly one word in D) and is a counterexample to the recent Uniformity Conjecture of Weichsel.     

Exploiting orbits in symmetric ILP

This paper describes components of a branch-and-cut algorithm for solving integer linear programs having a large symmetry group. It describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group. Pruning and orbit computations are performed by backtracking procedures using a Schreier-Sims table for representing the symmetry group. Applications to hard set covering problems, generation of covering designs and error correcting codes are given.Mathematics Subject Classification (2000):90C10, 90C27, 90C57     

The main involutions of the metaplectic group

We determine the set of automorphisms of a metaplectic group which lift the main involution of the general linear group over an infinite field. Some basic properties of these automorphisms are also established.

     

McKay quivers and absolute n-complete algebras

In this paper, we show that the McKay quiver of a finite subgroup of a general linear group is a regular covering of the McKay quiver of its intersection with the special linear group. Using this and our results on “returning arrows” in McKay quiver, we give an algorithm to construct the McKay quiver of a finite abelian group. Using this construction, we show how the cone and cylinder of an (n?1)-Auslander absolute n-complete algebra are truncated from the McKay quivers of abelian groups.     

Equipartition of interval partitions and an application to number theory

We give wider application and simpler proofs of results describing the rate at which the digits of one number-theoretic expansion determine those of another. The proofs are based on general measure-theoretic covering arguments and not on the dynamics of specific maps.

     

On the Picard group of a compact flat projective variety

In this note, we describe the Picard group of the class of compact, smooth, flat, projective varieties. In view of Charlap's work and Johnson's characterization, we construct line bundles over such manifolds as the holonomy-invariant elements of the Neron-Severi group of a projective flat torus covering the manifold. We prove a generalized version of the Appell-Humbert theorem which shows that the nontrivial elements of the Picard group are precisely those coming from the above construction. Our calculations finally give an estimate for the set of positive line bundles for such varieties.

     

Covering Sato-Levine invariants

Two covering versions of the Sato-Levine invariant are constructed which provide obstructions to certain two-component oriented links in the 3-sphere being link concordant to boundary links. These covering invariants are rational functions one of which detects both nonamphicheirality and noninvertibility of oriented links.

     

Lusin sets

We show that a set of real numbers is a Lusin set if, and only if, it has a covering property similar to the familiar property of Rothberger

     

Affine mappings of invertible operators

The infinite-dimensional analogues of the classical general linear group appear as groups of invertible elements of Banach algebras. Mappings of these groups onto themselves that extend to affine mappings of the ambient Banach algebra are shown to be linear exactly when the Banach algebra is semi-simple. The form of such linear mappings is studied when the Banach algebra is a C*-algebra.

     

A remark on analytic continuation

A simple example is given to show that the space of germs obtained by analytic continuation of a given germ need not be a covering space in the topological sense.

     

Pruning by isomorphism in branch-and-cut

 The paper presents a branch-and-cut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non-standard, cutting integer feasible solutions but leaving the optimal value of the problem unchanged. Pruning and cut generation are performed by backtracking procedures using a Schreier-Sims table for representing the group. Applications to hard set covering problems and to the generation of covering designs and error correcting codes are presented. Received: August 2001 / Accepted: October 2002 Publication online: December 9, 2002 Key Words. branch-and-cut – isomorphism pruning – symmetry     

Petersen图的点传递正则覆盖图

运用基图自同构能被提升的线性准则 ,对满足 :1覆叠变换群 K =Znp,2覆盖图的保簇变换群是点传递的 Petersen图的连通正则覆盖图进行了完全分类 .这种图共有 1 2种类型 .     

Geometric groups. I

We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.