Combinatorial aspects of extensions of Kronecker modules

Let kK be the path algebra of the Kronecker quiver and consider the category of finite dimensional right modules over kK (called Kronecker modules). We prove that extensions of Kronecker modules are field independent up to Segre classes, so they can be described purely combinatorially. We use in the proof explicit descriptions of particular extensions and a variant of the well known Green formula for Ringel-Hall numbers, valid over arbitrary fields. We end the paper with some results on extensions of preinjective Kronecker modules, involving the dominance ordering from partition combinatorics and its various generalizations.Comment: 11 page

Preinjective subfactors of preinjective Kronecker modules

Using a representation theoretical approach we give an explicit numerical characterization in terms of Kronecker invariants of the subfactor relation between two preinjective (and dually preprojective) Kronecker modules, describing explicitly a so called linking module as well. Preinjective (preprojective) Kronecker modules correspond to matrix pencils having only minimal indices for columns (respectively for rows). Thus our result gives a solution to the subpencil problem in these cases (including the completion), moreover the required computations are straightforward and can be carried out easily (both for checking the subpencil relation and constructing the completion pencils based on the linking module). We showcase our method by carrying out the computations on an explicit example.Comment: 13 page

The indecomposable preprojective and preinjective representations of the quiver ~D_n

Consider the quiver ~D_n and its finite dimensional representations over the field k. We know due to Ringel in that indecomposable representations without self extensions (called exceptional representations) can be exhibited using matrices involving as coefficients only 0 and 1, such that the number of nonzero coefficients is precisely d-1, where d is the global dimension of the representation. This means that the corresponding ''coefficient quiver'' is a tree, so we will call such a presentation a ''tree presentation''. In this paper we describe explicit tree presentations for the indecomposable preprojective and preinjective representations of the quiver ~D_n. In this way we generalize results obtained by Mr\' oz for the quiver ~D_4 and by Lorinczi and Szanto in for the quiver ~D_5

Hall polynomials and the Gabriel–Roiter submodules of simple homogeneous modules

Let k be an arbitrary field and Q be an acyclic quiver of tame type (that is, of type ˜ An, ˜Dn, ˜E6, ˜E7, ˜E8). Consider the path algebra kQ, the category of finite-dimensional right modules mod-kQ, and the minimal positive imaginary root of Q, denoted by δ. In the first part of the paper, we deduce that the Gabriel–Roiter (GR) inclusions in preprojective indecomposables and homogeneous modules of dimension δ, as well as their GR measures are field independent (a similar result due to Ringel being true in general over Dynkin quivers). Using this result, we can prove in a more general setting a theorem by Bo Chen which states that the GR submodule P of a homogeneous module R of dimension δ is preprojective of defect −1 and so the pair (R/P, P) is a Kronecker pair. The generalization consists in considering the originally missing case ˜E8 and using arbitrary fields (instead of algebraically closed ones). Our proof is based on the idea of Ringel (used in the Dynkin quiver context) of comparing all possible Hall polynomials with the special form they take in case of a GR inclusion. For this purpose, we determine (with the help of a program written in GAP) a list of tame Hall polynomials which may have further interesting applications

Stable project allocation under distributional constraints

In a two-sided matching market when agents on both sides have preferences the stability of the solution is typically the most important requirement. However, we may also face some distributional constraints with regard to the minimum number of assignees or the distribution of the assignees according to their types. These two kind of requirements can be challenging to reconcile in practice. Our research is motivated by two real applications, a project allocation problem and a workshop assignment problem, both involving some distributional constraints. We used integer programming techniques to find reasonably good solutions with regard to the stability and the distributional constraints. Our approach can be useful in a variety of different applications, such as resident allocation with lower quotas, controlled school choice or college admissions with affirmative action

Stable project allocation under distributional constraints

In a two-sided matching market when agents on both sides have preferences the stability of the solution is typically the most important requirement. However, we may also face some distributional constraints with regard to the minimum number of assignees or the distribution of the assignees according to their types. These two kind of requirements can be challenging to reconcile in practice. In this paper we describe two real applications, a project allocation problem and a workshop assignment problem, both involving some distributional constraints. We used integer programming techniques to find reasonably good solutions with regard to the stability and the distributional constraints. Our approach can be useful in a variety of different applications, such as resident allocation with lower quotas, controlled school choice or college admissions with addirmative action