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

Oddification of the cohomology of type A Springer varieties

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q=-1. This allows us to define graded modules over the Hecke algebra at q=-1 that are `odd' analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient of the skew polynomial ring by the left ideal of nonconstant odd symmetric functions. The top degree component of the odd cohomology of Springer varieties is identified with the corresponding Specht module of the Hecke algebra at q=-1.Comment: 21 pages, 2 eps file

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.

The sl_3 web algebra

In this paper we use Kuperberg’s sl3-webs and Khovanov’s sl3-foams to define a new algebra KS, which we call the sl3-web algebra. It is the sl3 analogue of Khovanov’s arc algebra. We prove that KS is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that KS is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K0 (WS )Q(q) , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that KS is a graded cellular algebra.info:eu-repo/semantics/publishedVersio

Translation functors and decomposition numbers for the periplectic Lie superalgebra p(n)\mathfrak{p}(n)

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of $\mathcal{F}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free

Translation functors and decomposition numbers for the periplectic Lie superalgebra p(n)\mathfrak{p}(n)

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of $\mathcal{F}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free

Translation functors and decomposition numbers for the periplectic Lie superalgebra p(n)\mathfrak{p}(n)

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of $\mathcal{F}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free

Connected orbits in topological generalized quadrangles

We study generalized quadrangles. After an investigation of the subgeometries that are generated by arbitrary sets of vertices, we consider orbits of connected subgroups of the automorphism group of topological generalized quadrangles. We deal with the problem of how a set of vertices has to be chosen in order that the union of the orbits generates a subquadrangle, or even the whole quadrangle. (orig.)Available from TIB Hannover: RN 2394(1730) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman