Cluster algebras of Grassmannians are locally acyclic
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- by Greg Muller and David E. Speyer PDF
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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.References
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Additional Information
- Greg Muller
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 905875
- David E. Speyer
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 663211
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3267-3281
- MSC (2010): Primary 13F60, 14M15
- DOI: https://doi.org/10.1090/proc/13023
- MathSciNet review: 3503695