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Weak separation, pure domains and cluster distance
Name
29_2018_394_ReferencePDF.pdf
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422.4 KB
Format
Adobe PDF
Checksum (MD5)
5ab6007de99552e946c40302559739e3
Author(s)
Farber, Miriam
Galashin, Pavel
Date Issued
February 1, 2018
Publisher
Springer International Publishing
Version
Author's final manuscript
Abstract
Abstract
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.
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DOI of Published Version
https://doi.org/10.1007/s00029-018-0394-2
