Degenerate flag varieties: moment graphs and Schr\"oder numbers

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schr\"oder number and the Poincar\'e polynomial is given by a natural statistics counting the number of diagonal steps in a Schr\"oder path. As an application we obtain a new combinatorial description of the large and small Schr\"oder numbers and their q-analogues.Comment: 25 page

An amphitropic cAMP-binding protein in yeast mitochondria

ABSTRACT: We describe the first example of a mitochondrial protein with a covalently attached phos-phatidylinositol moiety acting as a membrane anchor. The protein can be metabolically labeled with both stearic acid and inositol. The stearic acid label is removed by phospholipase D whereupon the protein with the retained inositol label is released from the membrane. This protein is a cAMP receptor of the yeast Saccharomyces cereuisiae and tightly associated with the inner mitochondrial membrane. However, it is converted into a soluble form during incubation of isolated mitochondria with Ca2+ and phospholipid (or lipid derivatives). This transition requires the action of a proteinaceous, N-ethylmaleimide-sensitive component of the intermembrane space and is accompanied by a decrease in the lipophilicity of the cAMP receptor. We propose that the component of the intermembrane space triggers the amphitropic behavior of the mitochondrial lipid-modified CAMP-binding protein through a phospholipase activity. Only in recent years specific fatty acids have been recog-nized to play important roles in the association of proteins with membranes. Both noncovalent and covalent interactions be-tween fatty acids and proteins have been reported. Among the latter are GTP-binding proteins (Molenaar et al., 1988)

On the uniqueness of promotion operators on tensor products of type A crystals

The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals.Comment: 31 pages; 8 figure

Order cones: A tool for deriving k-dimensional faces of cones of subfamilies of monotone games

In this paper we introduce the concept of order cone. This concept is inspired by the concept of order polytopes, a well-known object coming from Combinatorics. Similarly to order polytopes, order cones are a special type of polyhedral cones whose geometrical structure depends on the properties of a partially ordered set (brief poset). This allows to study these properties in terms of the subjacent poset, a problem that is usually simpler to solve. From the point of view of applicability, it can be seen that many cones appearing in the literature of monotone TU-games are order cones. Especially, it can be seen that the cones of monotone games with restricted cooperation are order cones, no matter the structure of the set of feasible coalitions