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dc.contributor.authorFarber, Miriam
dc.contributor.authorGalashin, Pavel
dc.description.abstractAbstract 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.en_US
dc.publisherSpringer International Publishingen_US
dc.rightsArticle 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.sourceSpringer International Publishingen_US
dc.titleWeak separation, pure domains and cluster distanceen_US
dc.eprint.versionAuthor's final manuscripten_US
dc.rights.holderSpringer International Publishing AG, part of Springer Nature
mit.metadata.statusAuthority Work and Publication Information Needed

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