×

Matroids from hypersimplex splits. (English) Zbl 1366.05024

Summary: A class of matroids is introduced which is very large as it strictly contains all paving matroids as special cases. As their key feature these split matroids can be studied via techniques from polyhedral geometry. It turns out that the structural properties of the split matroids can be exploited to obtain new results in tropical geometry, especially on the rays of the tropical Grassmannians.

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
14T99 Tropical geometry
14M15 Grassmannians, Schubert varieties, flag manifolds

Software:

polymake; TropLi
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bansal, N.; Pendavingh, R. A.; van der Pol, J. G., On the number of matroids, Combinatorica, 35, 3, 253-277 (2015) · Zbl 1363.05005
[2] Bonin, J. E.; de Mier, A., The lattice of cyclic flats of a matroid, Ann. Comb., 12, 2, 155-170 (2008) · Zbl 1145.05015
[3] Brouwer, A.; Shearer, J. B.; Sloane, N.; Smith, W. D., A new table of constant weight codes, IEEE Trans. Inform. Theory, 36, 6, 1334-1380 (1990) · Zbl 0713.94017
[4] Chatelain, V.; Ramírez Alfonsín, J. L., Matroid base polytope decomposition II: Sequences of hyperplane splits, Adv. Appl. Math., 54, 121-136 (2014) · Zbl 1284.05050
[5] De Loera, J. A.; Rambau, J.; Santos, F., Triangulations. Structures for Algorithms and Applications, Algorithms and Computation in Mathematics, vol. 25 (2010), Springer-Verlag: Springer-Verlag Berlin · Zbl 1207.52002
[6] Dress, A. W.M.; Wenzel, W., Valuated matroids, Adv. Math., 93, 2, 214-250 (1992) · Zbl 0754.05027
[7] Dukes, M., On the number of matroids on a finite set, Sém. Lothar. Combin., 51, Article B51g pp. (2004), 12pp · Zbl 1061.05020
[8] Edmonds, J., Submodular functions, matroids, and certain polyhedra, (Combinatorial Structures and Their Applications (Proc. Calgary Internat. Conf.). Combinatorial Structures and Their Applications (Proc. Calgary Internat. Conf.), Calgary, Alta., 1969 (1970), Gordon & Breach: Gordon & Breach New York), 69-87 · Zbl 0268.05019
[9] Feichtner, E. M.; Sturmfels, B., Matroid polytopes, nested sets and Bergman fans, Port. Math., 62, 4, 437-468 (2005) · Zbl 1092.52006
[10] Fife, T.; Oxley, J., Laminar matroids, European J. Combin., 62, 206-216 (2017) · Zbl 1358.05048
[11] Fink, A.; Rincón, F., Stiefel tropical linear spaces, J. Combin. Theory Ser. A, 135, 291-331 (2015) · Zbl 1321.15044
[12] Fujishige, S., A characterization of faces of the base polyhedron associated with a submodular system, J. Oper. Res. Soc. Japan, 27, 2, 112-129 (1984) · Zbl 0543.52008
[13] Gawrilow, E.; Joswig, M., : a framework for analyzing convex polytopes, (Polytopes - Combinatorics and Computation. Polytopes - Combinatorics and Computation, Oberwolfach, 1997. Polytopes - Combinatorics and Computation. Polytopes - Combinatorics and Computation, Oberwolfach, 1997, DMV Sem., vol. 29 (2000), Birkhäuser: Birkhäuser Basel), 43-73 · Zbl 0960.68182
[14] Gel’fand, I. M.; Goresky, M.; MacPherson, R. D.; Serganova, V. V., Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. Math., 63, 3, 301-316 (1987) · Zbl 0622.57014
[15] Herrmann, S., On the facets of the secondary polytope, J. Combin. Theory Ser. A, 118, 2, 425-447 (2011) · Zbl 1228.52008
[16] Herrmann, S.; Joswig, M., Splitting polytopes, Münster J. Math., 1, 109-141 (2008) · Zbl 1159.52016
[17] Herrmann, S.; Jensen, A.; Joswig, M.; Sturmfels, B., How to draw tropical planes, Electron. J. Combin., 16, 2, Article 6 pp. (2009), 26pp. Special volume in honor of Anders Björner · Zbl 1195.14080
[18] Herrmann, S.; Joswig, M.; Speyer, D., Dressians, tropical Grassmannians and their rays, Forum Math., 26, 6, 389-411 (2012)
[19] Hirai, H., A geometric study of the split decomposition, Discrete Comput. Geom., 36, 2, 331-361 (2006) · Zbl 1132.90012
[20] Knuth, D. E., The asymptotic number of geometries, J. Combin. Theory Ser. A, 16, 398-400 (1974) · Zbl 0278.05010
[21] Maclagan, D.; Sturmfels, B., Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161 (2015), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1321.14048
[22] Matsumoto, Y.; Moriyama, S.; Imai, H.; Bremner, D., Matroid enumeration for incidence geometry, Discrete Comput. Geom., 47, 1, 17-43 (2012) · Zbl 1236.05055
[23] Mayhew, D.; Newman, M.; Welsh, D.; Whittle, G., On the asymptotic proportion of connected matroids, European J. Combin., 32, 6, 882-890 (2011) · Zbl 1244.05047
[24] Murota, K., Discrete Convex Analysis, Monographs on Discrete Mathematics and Applications, vol. 10 (2003), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 1029.90055
[25] Oxley, J., Matroid Theory, Oxford Graduate Texts in Mathematics, vol. 21 (2011), Oxford University Press: Oxford University Press Oxford · Zbl 1254.05002
[26] Rincón, F., Local tropical linear spaces, Discrete Comput. Geom., 50, 3, 700-713 (2013) · Zbl 1281.14049
[27] Speyer, D. E., Tropical geometry (2005), University of California: University of California Berkeley, ProQuest LLC, Ann Arbor, MI, thesis (Ph.D.)
[28] Speyer, D. E., A matroid invariant via the \(K\)-theory of the Grassmannian, Adv. Math., 221, 3, 882-913 (2009) · Zbl 1222.14131
[29] Speyer, D.; Sturmfels, B., The tropical Grassmannian, Adv. Geom., 4, 3, 389-411 (2004) · Zbl 1065.14071
[30] (White, N., Theory of Matroids. Theory of Matroids, Encyclopedia of Mathematics and Its Applications, vol. 26 (1986), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0579.00001
[31] Ziegler, G. M., Lectures on 0/1-polytopes, (Polytopes - Combinatorics and Computation. Polytopes - Combinatorics and Computation, Oberwolfach, 1997. Polytopes - Combinatorics and Computation. Polytopes - Combinatorics and Computation, Oberwolfach, 1997, DMV Sem., vol. 29 (2000), Birkhäuser: Birkhäuser Basel), 1-41 · Zbl 0966.52012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.