Christof Geiß: Auslander algebras and initial seeds for cluster algebras

Let Q be a Dynkin quiver and Π the corresponding set of positive roots. For the preprojective algebra Λ associated to Q we produce a rigid Λ-module I_{Q} with r = |Π| pairwise non-isomorphic indecomposable summands by pushing the injective modules of the Auslander algebra of kQ to Λ.

If N is a maximal unipotent subgroup of a complex simply connected simple Lie group of type |Q|, then the coordinate ring ℂ[N] is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable summands of I_{Q} coincide with certain generalized minors which form an initial cluster for ℂ[N], and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of End_{Λ}(I_{Q}).