We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of selfinjective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of non-linear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations. We extend our construction to include recurrences with parameters, giving an explanation of some observations made by Gale. Finally, we point out a connection between quivers which arise in our classification and those arising in the context of quiver gauge theories.Comment: The final publication is available at www.springerlink.com. 42 pages, 35 figure

For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (set-theoretical) cross-section φ : G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w 0 in the Weyl group W. It assigns to any gB a representative g ∈ G together with a factorization into simple root subgroups and simple reflections. The cross-section φ is continuous along the components of Deodhar's decomposition of G/B. We introduce a generalization of the Chamber Ansatz and give formulas for the factors of g = φ(gB). These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety (G/B) ≥0 defined by Lusztig, giving a new proof of Lusztig's conjectured cell decomposition of (G/B) ≥0. We also give minimal sets of inequalities describing these cells.

We study Le Potier's strange duality conjecture on P 2 . We focus on the strange duality map SD c r n ,d which involves the moduli space of rank r sheaves with trivial first Chern class and second Chern class n, and the moduli space of 1-dimensional sheaves with determinant O P 2 (d) and Euler characteristic 0. By using tools in quiver representation theory, we show that SD c r n ,d is an isomorphisms for r = n or r = n − 1 or d ≤ 3, and in general SD c r n ,d is injective for any n ≥ r > 0 and d > 0.

Abstract. We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to