Clusters and seeds in acyclic cluster algebras
  We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky.We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra. 
Aslak Bakke Buan, Bethany Marsh, Idun Reiten, Gordana Todorov