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 dc.contributor.author Farber, Miriam dc.contributor.author Galashin, Pavel dc.date.accessioned 2021-09-20T17:20:25Z dc.date.available 2021-09-20T17:20:25Z dc.date.issued 2018-02-01 dc.identifier.uri https://hdl.handle.net/1721.1/131564 dc.description.abstract Abstract en_US Following the proof of the purity conjecture for weakly separated collections, recent years have revealed a variety of wider examples of purity in different settings. In this paper we consider the collection $$\mathcal A_{I,J}$$ A I , J of sets that are weakly separated from two fixed sets I and J. We show that all maximal by inclusion weakly separated collections $$\mathcal W\subset \mathcal A_{I,J}$$ W ⊂ A I , J are also maximal by size, provided that I and J are sufficiently “generic”. We also give a simple formula for the cardinality of $$\mathcal W$$ W in terms of I and J. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables. dc.publisher Springer International Publishing en_US dc.relation.isversionof https://doi.org/10.1007/s00029-018-0394-2 en_US dc.rights Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. en_US dc.source Springer International Publishing en_US dc.title Weak separation, pure domains and cluster distance en_US dc.type Article en_US dc.eprint.version Author's final manuscript en_US dc.type.uri http://purl.org/eprint/type/JournalArticle en_US eprint.status http://purl.org/eprint/status/PeerReviewed en_US dc.date.updated 2020-09-24T21:10:47Z dc.language.rfc3066 en dc.rights.holder Springer International Publishing AG, part of Springer Nature dspace.embargo.terms Y dspace.date.submission 2020-09-24T21:10:47Z mit.license PUBLISHER_POLICY mit.metadata.status Authority Work and Publication Information Needed
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