Karin Baur

A geometric description of m-cluster categories

We show that the m-cluster category of type An 1 is equivalent to a certain geometrically-defined category of diagonals of a regular nm + 2-gon. This generalises a result of Caldero, Chapoton and Schiffler f or m = 1. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the mth power of a translation quiver and show how it can be used to realise the m-cluster category in terms of the cluster category.

A Geometric Description of the m-cluster Categories of Type Dn

We show that the m-cluster category of type D_n is equivalent to a certain geometrically-defined category of arcs in a punctured regular nm-m+1-gon. This generalises a result of Schiffler for m=1. We use the notion of the mth power of a translation quiver to realise the m-cluster category in terms of the cluster category.

Nice parabolic subalgebras of reductive Lie algebras

This paper gives a classification of parabolic subalgebras of simple Lie algebras over C that are complexifications of parabolic subalgebras of real forms for which Lynch’s vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three dimensional simple Lie algebra).

Dimer models and cluster categories of Grassmannians

We associate a dimer algebra A to a Postnikov diagram D (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian (Gr) (k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay module T over the algebra B used to categorify the cluster structure of (Gr) (k,n) by Jensen-King-Su. It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disc. The construction and proof uses an interpretation of the diagram D, with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.

Secant dimensions of minimal orbits: computations and conjectures

We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits we give a short proof of the relation—known from the work of Ehrenborg, Catalisano–Geramita–Gimigliano, and Sturmfels–Sullivant—between the existence of certain codes and nondefectiveness of certain higher secant varieties.

Infinite friezes

We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a geometric interpretation of all entries of infinite friezes via matching numbers.

A geometric model of tube categories

Abstract We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented quiver of type A ∞ ∞ .

Categorification of a frieze pattern determinant

Broline, Crowe and Isaacs have computed the determinant of a matrix associated to a Conway-Coxeter frieze pattern. We generalise their result to the corresponding frieze pattern of cluster variables arising from the Fomin-Zelevinsky cluster algebra of type A. We give a representation-theoretic interpretation of this result in terms of certain configurations of indecomposable objects in the root category of type A.

A NORMAL FORM FOR ADMISSIBLE CHARACTERS IN THE SENSE OF LYNCH

Parabolic subalgebras p of semisimple Lie algebras define a Z-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of g on which the adjoint group of p acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In the case of Borel subalgebras a Richardson element of g1 is exactly one that involves all simple root spaces. It is however difficult to write down such nilpotent elements for general parabolic subalgebras. In this paper we give an explicit construction of admissible elements in g1 that uses as few root spaces as possible.

D-filtered modules and nilpotent orbits of a parabolic subgroup in On

We study certain ∆-filtered modules for the Auslander algebra of k[T ]/Tn ⋊ C2 where C2 is the cyclic group of order two. The motivation of this lies in the problem of describing the P -orbit structure for the action of a parabolic subgroup P of an orthogonal group. For any parabolic subgroup of an orthogonal group we construct a map from parabolic orbits to ∆-filtered modules and show that in the case of the Richardson orbit, the resulting module has no self-extensions.