150 research outputs found
Derived equivalences for symmetric groups and sl2- categorification
We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Brou´e’s abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also construct categorifications for category O of gln(C) and for rational representations of general linear groups over ¯Fp, where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard
Bounded derived categories of very simple manifolds
An unrepresentable cohomological functor of finite type of the bounded
derived category of coherent sheaves of a compact complex manifold of dimension
greater than one with no proper closed subvariety is given explicitly in
categorical terms. This is a partial generalization of an impressive result due
to Bondal and Van den Bergh.Comment: 11 pages one important references is added, proof of lemma 2.1 (2)
and many typos are correcte
Ultrastructural changes produced in plantlet leaves and protoplasts of Vitis vinifera cv. Cabernet Sauvignon by eutypine, a toxin from Eutypa lata
Eutypine is a toxin produced by Eutypa lata (Pers.: Pr.) Tul., the causal agent of dying-arm disease of the grapevine. Ultrastructural alterations induced by eutypine in leaf cells and protoplasts isolated from plantlets of Vitis vinifera cv. Cabernet Sauvignon, a very sensitive variety cultured in vitro, were observed for the first time by transmission electron microscopy. Eutypine caused early cytoplasm lysis and vesiculation followed by chloroplast swelling with thylakoid dilation. The eutypine-induced alterations of the cellular ultrastructure are similar to those previously described in vivo in the leaves of diseased grapevines. These results confirm that eutypine, synthesized by E. lata mycelium present in the trunk or arms, is involved in symptom expression of eutypiosis in the herbaceous parts of grapevine
Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
We construct an explicit isomorphism between blocks of cyclotomic Hecke
algebras and (sign-modified) Khovanov-Lauda algebras in type A. These
isomorphisms connect the categorification conjecture of Khovanov and Lauda to
Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally
graded, which allows us to exhibit a non-trivial Z-grading on blocks of
cyclotomic Hecke algebras, including symmetric groups in positive
characteristic.Comment: 32 pages; minor changes to section
Categorification of a linear algebra identity and factorization of Serre functors
We provide a categorical interpretation of a well-known identity from linear
algebra as an isomorphism of certain functors between triangulated categories
arising from finite dimensional algebras.
As a consequence, we deduce that the Serre functor of a finite dimensional
triangular algebra A has always a lift, up to shift, to a product of suitably
defined reflection functors in the category of perfect complexes over the
trivial extension algebra of A.Comment: 18 pages; Minor changes, references added, new Section 2.
Auslander-Buchweitz approximation theory for triangulated categories
We introduce and develop an analogous of the Auslander-Buchweitz
approximation theory (see \cite{AB}) in the context of triangulated categories,
by using a version of relative homology in this setting. We also prove several
results concerning relative homological algebra in a triangulated category
\T, which are based on the behavior of certain subcategories under finiteness
of resolutions and vanishing of Hom-spaces. For example: we establish the
existence of preenvelopes (and precovers) in certain triangulated subcategories
of \T. The results resemble various constructions and results of Auslander
and Buchweitz, and are concentrated in exploring the structure of a
triangulated category \T equipped with a pair (\X,\omega), where \X is
closed under extensions and is a weak-cogenerator in \X, usually
under additional conditions. This reduces, among other things, to the existence
of distinguished triangles enjoying special properties, and the behavior of
(suitably defined) (co)resolutions, projective or injective dimension of
objects of \T and the formation of orthogonal subcategories. Finally, some
relationships with the Rouquier's dimension in triangulated categories is
discussed.Comment: To appear at: Appl. Categor. Struct. (2011); 22 page
Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems
Most current methods for identifying coherent structures in
spatially-extended systems rely on prior information about the form which those
structures take. Here we present two new approaches to automatically filter the
changing configurations of spatial dynamical systems and extract coherent
structures. One, local sensitivity filtering, is a modification of the local
Lyapunov exponent approach suitable to cellular automata and other discrete
spatial systems. The other, local statistical complexity filtering, calculates
the amount of information needed for optimal prediction of the system's
behavior in the vicinity of a given point. By examining the changing
spatiotemporal distributions of these quantities, we can find the coherent
structures in a variety of pattern-forming cellular automata, without needing
to guess or postulate the form of that structure. We apply both filters to
elementary and cyclical cellular automata (ECA and CCA) and find that they
readily identify particles, domains and other more complicated structures. We
compare the results from ECA with earlier ones based upon the theory of formal
languages, and the results from CCA with a more traditional approach based on
an order parameter and free energy. While sensitivity and statistical
complexity are equally adept at uncovering structure, they are based on
different system properties (dynamical and probabilistic, respectively), and
provide complementary information.Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv
requirements; write first author for higher-resolution version
Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
This is the second of a series of four articles studying various
generalisations of Khovanov's diagram algebra. In this article we develop the
general theory of Khovanov's diagrammatically defined "projective functors" in
our setting. As an application, we give a direct proof of the fact that the
quasi-hereditary covers of generalised Khovanov algebras are Koszul.Comment: Minor changes, extra sections on Kostant modules and rigidity of cell
modules adde
Descent of Equivalences and Character Bijections
Categorical equivalences between block algebras of finite groups—such as Morita and derived equivalences—are well known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non-splitting fields. This article presents various results on the theme of descent to appropriate subfields and subrings. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centres in non-split situations and explain connections with Navarro’s generalisation of the Alperin–McKay conjecture. We show that Rouquier’s splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source
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