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 \(\mathcal A_{I,J}\) of sets that are weakly separated from two fixed sets I and J. We show that all maximal by inclusion weakly separated collections \(\mathcal W\subset \mathcal A_{I,J}\) are also maximal by size, provided that I and J are sufficiently “generic”. We also give a simple formula for the cardinality of \(\mathcal W\) 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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References
Danilov, V.I., Karzanov, A.V., Koshevoy, G.A.: Combined tilings and separated set-systems. Selecta Math. (N.S.) 23(2), 1175–1203 (2017)
Danilov, V.I., Karzanov, A.V., Koshevoy, G.A.: On maximal weakly separated set-systems. J. Algebraic Comb. 32(4), 497–531 (2010)
Farber, M., Postnikov, A.: Arrangements of equal minors in the positive Grassmannian. Adv. Math. 300, 788–834 (2016)
Galashin, P.: Plabic graphs and zonotopal tilings. arXiv:1611.00492 (2016)
Galashin, P., Postnikov, A.: Purity and separation for oriented matroids. arXiv:1708.01329 (2017)
Leclerc, B., Zelevinsky, A.: Quasicommuting families of quantum Plücker coordinates. In Kirillov’s Seminar on Representation Theory, Volume 181 of Amer. Math. Soc. Transl. Ser. 2, pp. 85–108. Amer. Math. Soc., Providence, RI (1998)
Oh, S.H., Speyer, D.E.: Links in the complex of weakly separated collections. J. Comb. 8(4), 581–592 (2017)
Oh, S., Postnikov, A., Speyer, D.E.: Weak separation and plabic graphs. Proc. Lond. Math. Soc. (3) 110(3), 721–754 (2015)
Postnikov, A.: Total positivity, Grassmannians, and networks. arXiv:math/0609764 (2006)
Scott, J.: Quasi-commuting families of quantum minors. J. Algebra 290(1), 204–220 (2005)
Scott, J.S.: Grassmannians and cluster algebras. Proc. Lond. Math. Soc. (3) 92(2), 345–380 (2006)
Acknowledgements
We thank Alex Postnikov for introducing us into the subject and a lot of fruitful conversations. We also thank the anonymous referee for their comments on the earlier versions of the manuscript.
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The work of M. Farber was supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.
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Farber, M., Galashin, P. Weak separation, pure domains and cluster distance. Sel. Math. New Ser. 24, 2093–2127 (2018). https://doi.org/10.1007/s00029-018-0394-2
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DOI: https://doi.org/10.1007/s00029-018-0394-2