Combinatorics and Algebra of Geometric Subdivision Operations

  • Fatemeh Mohammadi
  • Volkmar WelkerEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2176)


In the subsequent sections we survey results from combinatorics, discrete geometry and commutative algebra concerning invariants and properties of subdivisions of simplicial complexes. For most of the time we are interested in deriving results that hold for specific subdivision operations that are motivated from combinatorics, geometry and algebra. In particular, we study barycentric, edgewise and interval subdivisions (see Sect. 3 for the respective definitions). Even though we mention some suspicion that part of the results we present may only be a glimpse of what is true for general subdivision operations we do not focus on this aspect. In particular, we are quite sure that some asymptotic results and some convergence results from Sect. 9 are just instances of more general phenomena. Overall, retriangulations are subtle geometric operations and we refer the reader to the book De Loera et al. (Algorithms and Computation in Mathematics. Springer, Heidelberg, 2010) for a comprehensive introduction. Since our focus lies on specific constructions we make only little use of the theory from De Loera et al. (Algorithms and Computation in Mathematics. Springer, Heidelberg, 2010). Nevertheless, we are convinced that if one wants to go beyond specific subdivision operations it will become inevitable to dig deeper into the theory of triangulations.


Simplicial Complex Betti Number Monomial Ideal Free Resolution Eulerian Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 4.
    Aissen, M., Schoenberg, I.J., Whitney, A.: On generating functions of totally positive sequences I. J. Anal. Math. 2, 93–103 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 6.
    Athanasiadis, C.A.: Edgewise subdivisions, local h-polynomials and excedances in the wreath product \(\mathbb{Z}_{r} \wr \mathfrak{S}_{n}\). SIAM J. Disc. Math. 28, 1479–1492 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 7.
    Athanasiadis, C.A.: A survey of subdivisions and local h-vectors. In: The Mathematical Legacy of Richard P. Stanley, pp. 39–52. American Mathematical Society, Providence (2016)Google Scholar
  4. 9.
    Backelin, J.: On the rates of growth of the homologies of Veronese subrings. In: Algebra, Algebraic Topology, and Their Interactions (Stockholm, 1983). Lecture Notes in Mathematics. vol. 1183, pp. 79–100, Springer, Berlin (1986)Google Scholar
  5. 11.
    Batzies, E., Welker, V.: Discrete Morse theory for cellular resolutions. J. Reine Angew. Math. 543, 147–168 (2002)MathSciNetzbMATHGoogle Scholar
  6. 13.
    Bayer, D., Sturmfels, B.: Cellular resolutions of monomial modules. J. Reine Angew. Math. 502, 123–140 (1998)MathSciNetzbMATHGoogle Scholar
  7. 14.
    Bayer, D., Peeva, I., Sturmfels, B.: Cellular resolutions of monomial modules. Math. Res. Lett. 5, 31–46 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 15.
    Beck, M., Stapledon, A.: On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series. Math. Z. 264, 195–207 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 16.
    Bell, J., Skandera, M.: Multicomplexes and polynomials with real zeros, Discrete. Math. 307, 668–682 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 19.
    Billera, L.J., Björner, A.: Face numbers of polytopes and complexes. In: Handbook of Discrete and Computational Geometry. CRC Press Online - Series: Discrete Mathematics and Its Applications, pp. 291–310. CRC, Boca Raton (1997)Google Scholar
  11. 20.
    Billera, L.J., Lee, C.W.: Sufficiency of McMullen’s conditions for f-vectors of simplicial polytopes. Bull. Am. Math. Soc. New Series 2, 181–185 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 21.
    Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Am. Math. Soc. 260, 159–183 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 22.
    Boij, M., Migliore, J.C., Miró-Roig, R.M., Nagel, U., Zanello, F.: On the shape of a pure O-sequence. Mem. Am. Math. Soc. 218 (1024), viii+78 pp. (2012)Google Scholar
  14. 23.
    Bränden, P.: On linear transformations preserving the Pólya frequency property. Trans. Am. Math. Soc. 358, 3697–3716 (2006)CrossRefzbMATHGoogle Scholar
  15. 24.
    Brenti, F.: q-Eulerian polynomials arising from Coxeter groups. Eur. J. Combin. 15, 417–441 (1994)Google Scholar
  16. 25.
    Brenti, F., Welker, V.: f-vectors of barycentric subdivisions. Math. Z. 259, 849–865 (2008)Google Scholar
  17. 26.
    Brenti, F., Welker, V.: The Veronese construction for formal power series and graded algebras. Adv. Appl. Math. 42, 545–556 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 27.
    Brun, M., Römer, T.: Subdivisions of toric complexes. J. Algebraic Combin. 21, 423–448 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 29.
    Bruns, W., Herzog, J.: Cohen-Macaulay rings. In: Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)Google Scholar
  20. 31.
    Charney, R., Davis, M.: Euler characteristic of a nonpositively curved, piecewise Euclidean manifold. Pac. J. Math. 171, 117–137 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 32.
    Cheeger, J., Müller, W., Schrader, R.: On the curvature of piecewise flat spaces. Commun. Math. Phys. 92, 405–454 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 33.
    Comtet, L.: Advanced Combinatorics, Revised and Enlarged edition. D. Reidel, Dordrecht (1974)CrossRefzbMATHGoogle Scholar
  23. 38.
    Conca, A., Kubitzke, M., Welker, V.: Asymptotic syzygies of Stanley-Reisner rings of iterated subdivisions. Trans. Am. Math. Soc. (2017, to appear)Google Scholar
  24. 40.
    De Loera, J., Rambau, J., Santos, F.: Triangulations. Algorithms and Computation in Mathematics, vol. 25. Springer, Heidelberg (2010)Google Scholar
  25. 43.
    Delucchi, E., Pixton, A., Sabalka, L.: Face vectors of subdivided simplicial complexes. Discret. Math. 312, 248–257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 44.
    Diaconis, P., Fulman, J.: Carries, shuffling, and an amazing matrix. Am. Math. Mon. 116, 788–803 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 45.
    Diaconis, P., Fulman, J.: Carries, shuffling, and symmetric functions. Adv. Appl. Math. 43, 176–196 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 48.
    Edelsbrunner, H., Grayson, D.R.: Edgewise subdivision of a simplex. Discret. Comput. Geom. 24, 707–719 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 49.
    Ein, L., Lazarsfeld, R.: Asymptotic syzygies of algebraic varieties. Invent. Math. 190, 603–646 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 50.
    Ein, L., Erman, D., Lazarsfeld, R.: Asymptotics of random Betti tables. J. Reine Angew. Math. 702, 55–75 (2015)MathSciNetzbMATHGoogle Scholar
  31. 53.
    Eisenbud, D., Reeves, A., Totaro, B.: Initial ideals, Veronese subrings, and rates of algebras. Adv. Math. 109, 168–187 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 55.
    Freudenthal, H.: Simplizialzerlegung von beschränkter Flachheit. Ann. Math. 43, 580–582 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 56.
    Fröberg, R.: Determination of a class of Poincaré series. Math. Scand. 37, 29–39 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 58.
    Gal, S.R.: Real root conjecture fails for five and higher dimensional spheres. Discret. Comput. Geom. 34, 269–284 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 59.
    Gal, S.R., Januszkiewicz, T.: Odd-dimensional Charney-Davis conjecture. Discret. Comput. Geom. 44, 802–804 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 61.
    Grayson, D.R.: Exterior power operations in higher K-theory. K-Theory 3, 247–260 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 62.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at
  38. 63.
    Greene, C., Zaslavsky, T.: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. Trans. Am. Math. Soc. 280, 97–126 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 65.
    Holte, J.: Carries, combinatorics and an amazing matrix. Am. Math. Mon. 104, 138–149 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 71.
    Kubitzke, M., Murai, S.: Lefschetz properties and the Veronese construction. Math. Res. Lett. 19, 1043–1053 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 72.
    Kubitzke, M., Nevo, E.: The Lefschetz property for barycentric subdivisions of shellable complexes. Trans. Am. Math. Soc. 361, 6151–6163 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 75.
    Mermin, J.: The Eliahou-Kervaire resolution is cellular. J. Commut. Algebra 2, 55–78 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 76.
    Mohammadi, F.: Deformation of hyperplane arrangements, characteristic polynomials, and resolutions of powers of ideals. In preparation (2014)Google Scholar
  44. 77.
    Mohammadi, F.: Divisors on graphs, orientations, syzygies, and system reliability. J. Algebraic Combin. 43, 465–483 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 78.
    Mohammadi, F., Shokrieh, F.: Divisors on graphs, binomial and monomial ideals, and cellular resolutions. Math. Z. 283, 59–102 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 81.
    Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley Publishing Company, Menlo Park (1984)zbMATHGoogle Scholar
  47. 82.
    Munkres, J.R.: Topological results in combinatorics. Michigan Math. J. 31, 113–128 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 83.
    Nevo, E., Petersen, T.K., Tenner, B.E.: The γ-vector of a barycentric subdivision. J. Combin. Theory Ser. A 118, 1364–1380 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 85.
    Novik, I., Postnikov, A., Sturmfels, B.: Syzygies of oriented matroids. Duke Math. J. 111, 287–317 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 91.
    Peeva, I.: Graded syzygies. Algebra and Applications, vol. 14. Springer, London (2011)Google Scholar
  51. 93.
    Polishchuk, A.,  Positselski, L.: Quadratic Algebras. University Lecture Series, vol. 37. American Mathematical Society, Providence (2005)Google Scholar
  52. 94.
    Postnikov, A., Shapiro, B.: Trees, parking functions, syzygies, and deformations of monomial ideals. Trans. Am. Math. Soc. 356, 3109–3142 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 97.
    Reisner, G.A.: Cohen-Macaulay quotients of polynomial rings. Adv. Math. 21, 30–49 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 101.
    Savvidou, C.: Face numbers of cubical barycentric subdivisions. Preprint (2010). arXiv:1005.4156Google Scholar
  55. 103.
    Sommerville, D.: The relations connecting the angle sums and volume of a polytope in space of n dimensions. Proc. R. Soc. Ser. A 115, 103–119 (1927)CrossRefzbMATHGoogle Scholar
  56. 104.
    Stanley, R.P.: The number of faces of a simplicial convex polytope. Adv. Math. 35, 236–238 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 105.
    Stanley, R.P.: Subdivisions and local h-vectors. J. Am. Math. Soc. 5, 805–851 (1992)MathSciNetzbMATHGoogle Scholar
  58. 106.
    Stanley, R.P.: Flag f-vectors and the cd-index. Math. Z. 216, 483–499 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 107.
    Stanley, R.P.: Combinatorics and Commutative Algebra. Progress in Mathematics, 2nd edn, vol. 41. Birkhäuser, Boston (1996)Google Scholar
  60. 108.
    Stanley, R.P.: Enumerative Combinatorics, Volume 1. Cambridge Studies in Advanced Mathematics, 2nd edn, vol. 49. Cambridge University Press, Cambridge (2012)Google Scholar
  61. 112.
    Swartz, E.: Thirty five years and counting. Preprint (2014) arXiv:1411.0987Google Scholar
  62. 113.
    Taylor, D.: Ideals Generated by Monomials in an R-Sequence. Ph.D. Thesis, University of Chicago (1960)Google Scholar
  63. 114.
    Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, Oxford (1985)zbMATHGoogle Scholar
  64. 117.
    Walker, J.W.: Canonical homeomorphisms of posets. Eur. J. Combin. 9, 97–107 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 119.
    Zhou, X.: Effective non-vanishing of asymptotic adjoint syzygies. Proc. Am. Math. Soc. 142, 2255–2264 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Fachbereich Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany

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