Decomposition of the Witt-Burnside ring and Burnside ring of an abelian profinite group

Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers. The aim of this paper is to establish an isomorphism of functors , where denotes the q-deformed Witt-Burnside ring functor of G introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. To do this, we first establish an isomorphism of functors , where denotes the q-deformed Burnside ring functor of G which was also introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. As an application, we derive a pseudo-multiplicative property of the q-Möbius function associated to the lattice of open subgroups of the direct sum of G and H.     

Central idempotents of the bifree and left-free double Burnside ring

We determine the blocks, i.e., the primitive central idempotents, of the bifree double Burnside ring and the left-free double Burnside ring, as well as the primitive central idempotents of the algebras arising from scalar extension to ?.     

A Tate cohomology sequence for generalized Burnside rings

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G, then the mark morphism φ:D₊→D⁺ is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D⁺. As a consequence, we obtain an exact sequence of Mackey functors

     

Groups with isomorphic Burnside rings

We prove that for some families of finite groups, the isomorphism class of the group is completely determined by its Burnside ring. Namely, we prove the following: if two finite simple groups have isomorphic Burnside rings, then the groups are isomorphic; if G is either Hamiltonian or abelian or a minimal simple group, and G is any finite group such that B(G) B(G), then G G.Received: 22 April 2004     

Tornehave morphisms, II: The lifted Tornehave morphism and the dual of the Burnside functor

We introduce the lifted Tornehave morphism , an inflation Mackey morphism for finite groups, π being a set of primes, K the kernel of linearization, and B^∗ the dual of the Burnside functor. For p-groups, is unique up to scalar multiples. It induces two morphisms of biset functors, one with a codomain associated with a subgroup of the Dade group, the other with a codomain associated with a quotient of the Burnside unit group.     

Vanishing evaluations of simple functors

The classification of simple biset functors is known, but the evaluation of a simple biset functor at a finite group G$G$ may be zero. We investigate various situations where this happens, as well as cases where this does not occur. We also prove a closed formula for such an evaluation under some restrictive conditions on  G$G$.     

The Loewy length of a tensor product of modules of a dihedral two-group

While the finite-dimensional modules of the dihedral 2-groups over fields of characteristic 2 were classified over 30 years ago, very little is known about the tensor products of such modules. In this article, we compute the Loewy length of the tensor product of two modules of a dihedral 2-group in characteristic 2. As an immediate consequence, we determine when such a tensor product has a projective direct summand.     

The Ring of Modules with Endo-permutation Source

In this paper we consider certain subalgebras of the Green algebra (representation algebra) of a finite group G. One such algebra is spanned by the isomorphism classes of all indecomposable modules whose source is an endo-permutation module. This algebra can be embedded into a finite direct product of Laurent polynomial rings in finitely many variables over a field. Another such algebra is spanned by the isomorphism classes of all irreducibly generated modules. When G is p-solvable then this algebra is finite-dimensional and split semisimple.R. Boltje was supported by the NSF, DMS-0200592 and 0128969. B. Külshammer was supported by the DAAD.     

Locally strong endomorphisms of paths

We determine the number of locally strong endomorphisms of directed and undirected paths—direction here is in the sense of a bipartite graph from one partition set to the other. This is done by the investigation of congruence classes, leading to the concept of a complete folding, which is used to characterize locally strong endomorphisms of paths. A congruence belongs to a locally strong endomorphism if and only if the number l of congruence classes divides the length of the original path and the points of the path are folded completely into the l classes, starting from 0 to l and then back to 0, then again back to l and so on. It turns out that for paths locally strong endomorphisms form a monoid if and only if the length of the path is prime or equal to 4 in the undirected case and in the directed case also if the length is 8. Finally some algebraic properties of these monoids are described.     

The Lagrangian Gauss image of a compact surface in Minkowski 3-Space

A generic compact surfaceQ in Minkowski 3-Space is naturally stratified by the loci where the orthogonal line bundle is tangent to the next lower stratum,SP D ⁰ Q M³. To each component inD ⁰ we associate a light-like hypersurface and in turn a Lagrangian loop in the cotangent bundle of the circle. We then establish an inequality relating the Euler characteristic of the indefinite component ofQ with the total Gauß-Maslov index of the associated Lagrangian loops.     

On the continuity of separately continuous bihomomorphisms

Separately continuous bihomomorphisms on a product of convergence or topological groups occur with great frequency. Of course, in general, these need not be jointly continuous. In this paper, we exhibit some results of Banach-Steinhaus type and use these to derive joint continuity from separate continuity. The setting of convergence groups offers two advantages. First, the continuous convergence structure is a powerful tool in many duality arguments. Second, local compactness and first countability, the usual requirements for joint continuity, are available in much greater abundance for convergence groups.     

On a subalgebra of the centre of a group ring II

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By we denote the F-subspace of the centre of the group ring FG spanned by the p-regular conjugacy class sums. J. Murray proved that is an algebra, if G is a symmetric or alternating group. This can be used for the computation of the block idempotents of FG. We proved that is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Külshammer generalized this result to blocks with abelian defect groups. Here, we show that is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore and are algebras for all primes p and all prime powers q. Furthermore we prove that is an algebra for the simple Suzuki-groups Sz(q), where q is a certain power of 2 and p is an arbitrary prime dividing |Sz(q)|. Received: 18 May 2007     

Periodicity of Adams operations on the Green ring of a finite group

The Adams operations and on the Green ring of a group G over a field K arise from the study of the exterior powers and symmetric powers of KG-modules. When G is finite and K has prime characteristic p we show that and are periodic in n if and only if the Sylow p-subgroups of G are cyclic. In the case where G is a cyclic p-group we find the minimum periods and use recent work of Symonds to express in terms of .     

A commutative analogue of the group ring

Throughout, all rings R will be commutative with identity element. In this paper we introduce, for each finite group G, a commutative graded Z-algebra R_G. This classifies the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element.In the case when G is an elementary Abelian p-group it turns out that R_G is closely related to the symmetric algebra over F_p of the dual of G. We intend in subsequent papers to explore the close relationship between G and R_G in the case of a general (possibly non-Abelian) group G.Here we show that the Krull dimension of R_G is the maximal rank r of an elementary Abelian subgroup E of G unless either E is cyclic or for some such E its normalizer in G contains a non-trivial cyclic group which acts faithfully on E via “scalar multiplication” in which case it is r+1.     

The Green rings of pointed tensor categories of finite type

In this paper, we compute the Clebsch–Gordan formulae and the Green rings of connected pointed tensor categories of finite type.