Auslander algebras and initial seeds for cluster algebras

Let $Q$ be a Dynkin quiver and $\Pi$ the corresponding set of positive roots. For the preprojective algebra $\Lambda$ associated to $Q$ we produce a rigid $\Lambda$-module $I_Q$ with $r=|\Pi|$ pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of $kQ$ to $\Lambda$. 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 direct 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 $End_{\Lambda}(I_Q)$. Finally, we exploit the fact that the categories of injective modules over $\Lambda$ and over its covering $\tilde{\Lambda}$ are triangulated in order to show several interesting identities in the respective stable module categories.