On rectangular Kronecker coefficients

We show that rectangular Kronecker coefficients stabilize when the lengths of the sides of the rectangle grow, and we give an explicit formula for the limit values in terms of invariants of mathfraksl_nmathfrak{s}l_{n}.     

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ₁,…,λd²) of size ?d there exists k?1 such that the tensor square of the irreducible representation of the symmetric group Sk?d with respect to the rectangular partition (k?,…,k?) contains the irreducible representation corresponding to the stretched partition (kλ₁,…,kλd²). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.     

Twisted smooth Deligne cohomology

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful.     

Leopold Kronecker

Ohne ZusammenfassungAbgedruckt aus dem Jahresbericht der Deutschen Mathematiker-Vereinigung 2. Band, 1892.     

Integral Deligne cohomology for real varieties

We develop an integral version of Deligne cohomology for smooth proper real varieties. For this purpose the role played by singular cohomology in the complex case has to be replaced by the ordinary bigraded Gal(mathbbC/mathbbR){Gal(mathbb{C}/{mathbb{R}})}-equivariant cohomology of Lewis et al. (Bull Am Math Soc (N.S.) 4(2):208–212, 1981), the equivariant counterpart of singular cohomology. The theory is aimed at giving more precise information about the 2-primary components of regulators. We establish basic properties and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2.     

Higher order operations in Deligne cohomology

Oblatum 10-VII-1993 & 26-IX-1994     

Connectedness of Kronecker sum and partial Kronecker row sum of designs

In this paper we discuss connectedness of a design which is a Kronecker sum or a partial Kronecker row sum of any two equi-replicate and equi-block size designs.     

Kronecker products and BIBDs

Recursive constructions are given which permit, under conditions described in the paper, a (v, b, r, k, λ)-configuration to be used to obtain a (v′, b′, r′, k, λ)-configuration.     

Cohomology of small categories with coefficients in an Abelian category with exact products

Conclusion The spectral sequence of Theorem 2.1 is widely known for the case where the group P acts freely on a simplicial set. Corollary 3.2 shows that both the result of [11] and the known estimates of the cohomological dimension of a directed colimit of groups are incorporated in this spectral sequence.Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 4, pp. 210–215, July–August, 1989.     

A remark on a theorem by Deligne

We give a proof avoiding spectral sequences of Deligne's decomposition theorem for objects in a triangulated category admitting a Lefschetz homomorphism.

     

Exact categories and vector space categories

In a series of papers additive subbifunctors of the bifunctor are studied in order to establish a relative homology theory for an artin algebra . On the other hand, one may consider the elements of as short exact sequences. We observe that these exact sequences make into an exact category if and only if is closed in the sense of Butler and Horrocks.

Concerning the axioms for an exact category we refer to Gabriel and Roiter's book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories.

In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation.

Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor and in general do not preserve at all the usual exact structure of .

By showing the closedness of suitable subbifunctors of we can equip with an exact structure such that some reduction functors actually become `exact'. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them exist.

Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite.