The positive Dressian equals the positive tropical Grassmannian
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- by David Speyer and Lauren K. Williams;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 330-353
- DOI: https://doi.org/10.1090/btran/67
- Published electronically: April 9, 2021
- HTML | PDF
Abstract:
The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces – the positive Dressian, and the positive tropical Grassmannian (which we introduced roughly fifteen years ago in [J. Algebraic Combin. 22 (2005), pp. 189–210]) – so it is natural to ask how these two positive spaces compare. In this paper we show that the positive Dressian equals the positive tropical Grassmannian. Using the connection between the positive Dressian and regular positroidal subdivisions of the hypersimplex, we use our result to give a new “tropical” proof of da Silva’s 1987 conjecture (first proved in 2017 by Ardila-Rincón-Williams) that all positively oriented matroids are realizable. We also show that the finest regular positroidal subdivisions of the hypersimplex consist of series-parallel matroid polytopes, and achieve equality in Speyer’s $f$-vector theorem. Finally we give an example of a positroidal subdivision of the hypersimplex which is not regular, and make a connection to the theory of tropical hyperplane arrangements.References
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Bibliographic Information
- David Speyer
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
- MR Author ID: 663211
- Email: speyer@umich.edu
- Lauren K. Williams
- Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
- MR Author ID: 611667
- Email: williams@math.harvard.edu
- Received by editor(s): March 29, 2020
- Received by editor(s) in revised form: January 15, 2021
- Published electronically: April 9, 2021
- Additional Notes: The first author was partially supported by NSF grants DMS-1855135 and DMS-1854225. The second author was partially supported by NSF grants DMS-1854316 and DMS-1854512.
- © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 330-353
- MSC (2020): Primary 05E99; Secondary 14M15
- DOI: https://doi.org/10.1090/btran/67
- MathSciNet review: 4241765