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Cluster algebras of Grassmannians are locally acyclic

Authors: Greg Muller and David E. Speyer
Journal: Proc. Amer. Math. Soc. 144 (2016), 3267-3281
MSC (2010): Primary 13F60, 14M15
Published electronically: March 16, 2016
MathSciNet review: 3503695
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Abstract: 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 show the stronger result that all positroid varieties are locally acyclic. However, it has not been shown that all positroid varieties have a cluster structure in the expected manner, so what we actually prove is that certain cluster varieties associated to Postnikov's alternating strand diagrams are locally acylic. We actually establish a slightly stronger property than local acyclicity that is designed to facilitate proofs involving the Mayer-Vietores sequence.

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Additional Information

Greg Muller
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

David E. Speyer
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Received by editor(s): June 20, 2015
Received by editor(s) in revised form: October 4, 2015
Published electronically: March 16, 2016
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2016 American Mathematical Society

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