Cluster Algebras of Grassmannians are Locally Acyclic

By Greg Muller, David E. Speyer. 2014

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Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and most fundamental examples of a cluster algebra is the homogenous coordinate ring of the Grassmannian. We show that the Grassmannian is locally acyclic. Morally, we are in fact showing the stronger result that all positroid varieties are locally acyclic. However, it has not been shown that all positroid varieties have cluster structure, so what we actually prove is that certain cluster varieties associated to Postnikov's alternating strand diagrams are locally acylic. Moreover, we actually establish a slightly stronger property than local acyclicity, which we term the Louise property, that is designed to facilitate proofs involving the Mayer-Vietores sequence.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/188dbd0f018c37aab7d016a816eddf31eb28cb8e

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Citations in the corpus

Positroid varieties and cluster algebras (2019, background): “ r C[(q0 r ) 1] ! is equal to the upper cluster algebra A(Q D), dened to be the intersection of Laurent polynomial rings for all clusters of A(Q D). By Lemma 4.7, we have C[ v;w] (˝)(A(Q D)). By [MS16], A(Q D) is a locally acyclic cluster algebra, and by [Mul13, Theorem 4.1], we have A(Q D) = A(Q D”
The twist for positroid varieties (2016, background): “Assuming the cluster structure on Π̃◦(M) described in [22], there is a rational cluster ensemble map χ : Π̃◦(M) –→ X , where X is the associated X -cluster variety [7].”
Cluster structures on strata of flag varieties (2014, background): “nd v = wKu for some u ∈WK. In this case the strata RwKu,w ≃R K wKu,w, also called open positroid varieties, have been studied by many authors, see [P, BGY, KLS2, GLL1, GLL2]. In their recent preprint [MS], Muller and Speyer have conjectured that the coordinate rings of open positroid varieties are i”