In this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is encoded using (target labelings of) Postnikov's plabic graphs. This result generalizes a theorem of Scott [Proc. Lond. Math. Soc. (3) 92 (2006) 345-380] for the Grassmannian, and proves a folklore conjecture for Schubert varieties that has been believed by experts since Scott's result [Proc. Lond. Math. Soc. (3) 92 (2006) 345-380], though the statement was not formally written down until Muller-Speyer explicitly conjectured it [Proc. Lond. Math. Soc. (3) 115 (2017) 1014-1071]. To prove this conjecture we use a result of Leclerc [Adv. Math. 300 (2016) 190-228] who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety can be identified with cluster algebras. Our proof also uses a construction of Karpman [J. Combin. Theory Ser. A 142 (2016) 113-146] to build plabic graphs associated to reduced expressions. We additionally generalize our result to the setting of skew-Schubert varieties; the latter result uses generalized plabic graphs, that is, plabic graphs whose boundary vertices need not be labeled in cyclic order. Contents

Abstract. We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type A. We prove that such sequences have length n + t, where n is the number of vertices and t is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.

Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by the maximal green sequences of a quiver. We combine the combinatorics of surface triangulations and the basics of scattering diagrams to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of an annulus or a punctured disk.

We propose a new approach to study the relation between the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B = C E. This new approach consists of using the induction functor − ⊗ C B as well as the coinduction functor D(B ⊗ C D−). We give an explicit construction of injective resolutions of projective B-modules, and as a consequence, we obtain a new proof of the 1-Gorenstein property for cluster-tilted algebras. We show that DE is a partial tilting and a τ -rigid C-module and that the induced module DE ⊗ C B is a partial tilting and a τ -rigid B-module. Furthermore, if C = End A T for a tilting module T over a hereditary algebra A, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor Hom C A (T, −) from the cluster-category of A to the module category of B. We also study the question which B-modules are actually induced or coinduced from a module over a tilted algebra.

L-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each L-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of the corresponding positroid variety. We study quivers arising from these diagrams and show that they can be constructed from the well-behaved quivers associated to Grassmannians by deleting and merging certain vertices. Then, we prove that quivers coming from arbitrary L-diagrams, and more generally reduced plabic graphs, admit a particular sequence of mutations called a green-to-red sequence.