Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with arXiv:1405.5920 by other author

Transitive and Co-Transitive Caps

A cap in PG(r,q) is a set of points, no three of which are collinear. A cap is said to be transitive if its automorphism group in PGammaL(r+1,q) acts transtively on the cap, and co-transitive if the automorphism group acts transtively on the cap's complement in PG(r,q). Transitive, co-transitive caps are characterized as being one of: an elliptic quadric in PG(3,q); a Suzuki-Tits ovoid in PG(3,q); a hyperoval in PG(2,4); a cap of size 11 in PG(4,3); the complement of a hyperplane in PG(r,2); or a union of Singer orbits in PG(r,q) whose automorphism group comes from a subgroup of GammaL(1,q^{r+1}).Comment: To appear in The Bulletin of the Belgian Mathematical Society - Simon Stevi

Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory

We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group $( G , + )$, a set of weights $W \subset \mathbb{Z}$, and an integral parameter $m$, the $m$-wise Davenport constant with weights $W$ is the smallest integer $n$ such that each sequence over $G$ of length $n$ has at least $m$ disjoint zero-subsums with weights $W$. And, for an integral parameter $d$, the $d$-constrained Davenport constant with weights $W$ is the smallest $n$ such that each sequence over $G$ of length $n$ has a zero-subsum with weights $W$ of size at most $d$. First, we establish a link between these two types of constants and several basic and general results on them. Then, for elementary $p$-groups, establishing a link between our constants and the parameters of linear codes as well as the cardinality of cap sets in certain projective spaces, we obtain various explicit results on the values of these constants

On rational homology disk smoothings of valency 4 surface singularities

Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are the three classes whose resolution dual graph has central vertex with valency 4. We give a uniform "quotient construction" of the QHD smoothings for these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes in a previous paper; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different QHD smoothings for the first class. We also prove a general formula for the dimension of a QHD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that "most" H-shaped resolution graphs cannot be the graph of a singularity with a QHD smoothing. This result, plus recent work of Bhupal-Stipsicz, is evidence for a general Conjecture: The only complex surface singularities with a QHD smoothing are the (known) weighted homogeneous examples.Comment: 28 pages: title changed, typos fixed, references and small clarifications adde