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Valuations for Matroid Polytope Subdivisions

Published online by Cambridge University Press:  20 November 2018

Federico Ardila ,
Alex Fink  and
Felipe Rincón
Federico Ardila*
Affiliation:
San Francisco State University, San Francisco, CA, USA
Alex Fink*
Affiliation:
University of California, Berkeley, Berkeley, CA, USA
Felipe Rincón*
Affiliation:
Universidad de Los Andes, Bogotá, Colombia
*
federico@math.sfsu.edu
finka@math.berkeley.edu
ed-rinco@uniandes.edu.co
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Abstract

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We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.

Keywords

05B3552B4052B4552C22
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 6 , 14 December 2010 , pp. 1228 - 1245
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Ardila, F., The Catalan matroid. J. Combin. Theory Ser. A 104(2003), no. 1, 49–62. doi:10.1016/S0097-3165(03)00121-3Google Scholar
[2] Björner, A., Las Vergnas, M., Sturmfels, B., White, N., and Ziegler, G. M., Oriented matroids. Encyclopedia of Mathematics and its Applications, 46, Cambridge University Press, Cambridge, 1993.Google Scholar
[3] Billera, L. J., Jia, N., and Reiner, V., A quasisymmetric function for matroids. European J. Combin. 30(2009), no. 8, 1727–1757. doi:10.1016/j.ejc.2008.12.007Google Scholar
[4] Bonin, J. and de Mier, A., Lattice path matroids: structural properties. European J. Combin. 27(2006), no. 5, 701–738. doi:10.1016/j.ejc.2005.01.008Google Scholar
[5] Borovik, A., Gelfand, I., and N., White, Coxeter matroids. Progress in Mathematics, 216, Birkhäuser Boston, Boston, MA, 2003.Google Scholar
[6] Brylawski, T. and Oxley, J., The Tutte polynomial and its applications. In: Matroid applications, Encyclopedia Math. Appl., 40, Cambridge University Press, Cambridge, 1992, pp. 123–225.Google Scholar
[7] Crapo, H. H., Single-element extensions of matroids. J. Res. Nat. Bur. Standards Sect. B 69B(1965), 55–65.Google Scholar
[8] Crapo and W, H.. Schmitt, A free subalgebra of the algebra of matroids. European J. Combin. 26(2005), no. 7, 1066–1085. doi:10.1016/j.ejc.2004.05.006Google Scholar
[9] Derksen, H., Symmetric and quasi-symmetric functions associated to polymatroids. J. Algebraic Combin. 30(2009), no. 1, 43–86. doi:10.1007/s10801-008-0151-2Google Scholar
[10] Derksen, H. and Fink, A., Valuative invariants for polymatroids. Adv. Math., in press.Google Scholar
[11] Dress and W, A..Wenzel, Valuated matroids. Adv. Math. 93(1992), no. 2, 214–250. doi:10.1016/0001-8708(92)90028-JGoogle Scholar
[12] Gelfand, I., Goresky, R., Mac Pherson, R., and V. Serganova. Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. in Math. 63(1987), no. 3, 301–316. doi:10.1016/0001-8708(87)90059-4Google Scholar
[13] Hacking, P., Keel, S., and J. Tevelev. Compactification of the moduli space of hyperplane arrangements. J. Algebraic Geom. 15(2006), no. 4, 657–680.Google Scholar
[14] Kapranov, M. M., Chow quotients of Grassmannians I. In: I. M. Gel’fand Seminar, Adv. Soviet Math. 16, American Mathematical Society, Providence, RI, 1993, pp. 29–110.Google Scholar
[15] Klivans, C., Combinatorial properties of shifted complexes. Ph.D. Thesis, Massachusetts Institute of Technology, 2003.Google Scholar
[16] Lafforgue, L., Chirurgie des grassmanniennes. CR M Monograph Series, 19, American Mathematical Society, Providence, RI, 2003.Google Scholar
[17] Lafforgue, L., Pavages des simplexes, schémas de graphes recollés et compactification des PGLn+1 r /PGLr . Invent. Math. 136(1999), no. 1, 233–271. doi:10.1007/s002220050309Google Scholar
[18] Mc Mullen, P., Valuations and dissections. In: Handbook of convex geometry. North-Holland Publishing Co., Amsterdam, 1993, pp. 933–988.Google Scholar
[19] Oxley, J. G., Matroid theory. Oxford University Press, New York, 1992.Google Scholar
[20] Rota, G.-C., On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2(1964), 340–368. doi:10.1007/BF00531932Google Scholar
[21] Rudin, W., Functional analysis. Mc Graw-Hill Series in Higher Mathematics, Mc Graw-Hill, New York, 1973.Google Scholar
[22] Schrijver, A., Combinatorial optimization. Polyhedra and efficiency. Algorithms and Combinatorics Series, 24, Springer-Verlag, Berlin, 2003, Chpts. 70–83.Google Scholar
[23] Speyer, D., Tropical linear spaces. SIA M J. Discrete Math. 22(2008), no. 4, 1527–1558. doi:10.1137/080716219Google Scholar
[24] Speyer, D., A matroid invariant via the K-theory of the Grassmannian. Adv. Math. 221(2009), no. 3, 882–913. doi:10.1016/j.aim.2009.01.010Google Scholar
[25] Stanley, R. P., Enumerative combinatorics, vol. 1. Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 1997.Google Scholar