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Matching fields and lattice points of simplices

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Matching fields and lattice points of simplices. / Smith, Ben; Loho, Georg.
In: Advances in Mathematics, Vol. 370, 107232, 26.08.2020.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Smith, B & Loho, G 2020, 'Matching fields and lattice points of simplices', Advances in Mathematics, vol. 370, 107232. https://doi.org/10.1016/j.aim.2020.107232

APA

Smith, B., & Loho, G. (2020). Matching fields and lattice points of simplices. Advances in Mathematics, 370, Article 107232. https://doi.org/10.1016/j.aim.2020.107232

Vancouver

Smith B, Loho G. Matching fields and lattice points of simplices. Advances in Mathematics. 2020 Aug 26;370:107232. doi: 10.1016/j.aim.2020.107232

Author

Smith, Ben ; Loho, Georg. / Matching fields and lattice points of simplices. In: Advances in Mathematics. 2020 ; Vol. 370.

Bibtex

@article{21e9a964b3b0454cab2595f310090d01,
title = "Matching fields and lattice points of simplices",
abstract = "We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) [26] on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex.Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.",
keywords = "Bipartite graph, Lattice points, Linkage property, Matching field, Product of simplices, Triangulation",
author = "Ben Smith and Georg Loho",
year = "2020",
month = aug,
day = "26",
doi = "10.1016/j.aim.2020.107232",
language = "English",
volume = "370",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

RIS

TY - JOUR

T1 - Matching fields and lattice points of simplices

AU - Smith, Ben

AU - Loho, Georg

PY - 2020/8/26

Y1 - 2020/8/26

N2 - We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) [26] on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex.Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.

AB - We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) [26] on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex.Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.

KW - Bipartite graph

KW - Lattice points

KW - Linkage property

KW - Matching field

KW - Product of simplices

KW - Triangulation

U2 - 10.1016/j.aim.2020.107232

DO - 10.1016/j.aim.2020.107232

M3 - Journal article

VL - 370

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107232

ER -