We show that the m-cluster category of type A n−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 for 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… (More)
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… (More)
We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type A. The formula involves various aspects of combinatorics of Dyck paths and leads to a new interesting integral sequence.
We show that the m-cluster category of type Dn 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.
abstract 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 also point out a relation between the existence of certain codes and… (More)
Parabolic subalgebras of semi-simple Lie algebras decompose as p = m ⊕ n where m is a Levi factor and n the corresponding nilradical. By Richardsons theorem [R], there exists an open orbit under the action of the adjoint group P on the nilradical. The elements of this dense orbits are known as Richardson elements. In this paper we describe a normal form for… (More)
We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all , as well as for the variety F of incident point-line pairs in P 2. For P 2 × P 1 and F the results are new, while the proofs for the other… (More)
We construct frieze patterns of type D N with entries which are numbers of matchings between vertices and triangles of corresponding trian-gulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type D N , we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in… (More)
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… (More)