The local \(h\)-polynomial of the edgewise subdivision of the simplex. (English) Zbl 1425.05163

Summary: The \(r\)-fold edgewise subdivision is a well studied flag triangulation of the simplex with interesting algebraic, combinatorial and geometric properties. An important enumerative invariant, namely the local \(h\)-polynomial, of this triangulation is computed and shown to be \(\gamma\)-nonnegative by providing explicit combinatorial interpretations to the corresponding coefficients. A construction of a flag triangulation of the seven-dimensional simplex whose local \(h\)-polynomial is not real-rooted is also described.


05E45 Combinatorial aspects of simplicial complexes
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
55U10 Simplicial sets and complexes in algebraic topology
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[1] C.A. Athanasiadis, Flag subdivisions and γ-vectors, Pacific J. Math. 259 (2012), 257-278. · Zbl 1255.05200
[2] C.A. Athanasiadis, Edgewise subdivisions, local h-polynomials and excedances in the wreath product Zro Sn, SIAM J. Discrete Math. 28 (2014), 1479-1492. · Zbl 1305.05232
[3] C.A. Athanasiadis, A survey of subdivisions and local h-vectors, in The Mathematical Legacy of Richard P. Stanley (P. Hersh, T. Lam, P. Pylyavskyy and V. Reiner, eds.), Amer. Math. Soc. (to appear).
[4] A. Bj¨orner, Topological methods, in Handbook of Combinatorics (R.L. Graham, M. Gr¨otschel and L. Lov´asz, eds.), North Holland, Amsterdam, 1995, pp. 1819-1872.
[5] M. B¨okstedt, W.C. Hsiang and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 11 (1993), 465-539. · Zbl 0804.55004
[6] F. Brenti and V. Welker, The Veronese construction for formal power series and graded algebras, Adv. in Appl. Math. 42 (2009), 545-556. · Zbl 1230.05299
[7] M. Brun and T. R¨omer, Subdivisions of toric complexes, J. Algebraic Combin. 21 (2005), 423-448. · Zbl 1080.14058
[8] A. Conca, M. Juhnke-Kubitzke and V. Welker, Asymptotic syzygies of Stanley-Reisner rings of iterated subdivisions, arXiv:1411.3695. · Zbl 1386.13059
[9] J.A. De Loera, J. Rambau and F. Santos, Triangulations: Structures for Algorithms and Applications, Algorithms and Computation in Mathematics 25, Springer, 2010. · Zbl 1207.52002
[10] H. Edelsbrunner and D.R. Grayson, Edgewise subdivision of a simplex, Discrete Comput. Geom. 24 (2000), 707-719. · Zbl 0968.51016
[11] H. Freudenthal, Simplizialzerlegung von beschr¨ankter Flachheit, Ann. of Math. 43 (1942), 580-582.
[12] S.R. Gal, Real root conjecture fails for five- and higher-dimensional spheres, Discrete Comput. Geom. 34 (2005), 269-284. · Zbl 1085.52005
[13] D.R. Grayson, Exterior power operations on higher K-theory, K Theory 3 (1989), 247-260. · Zbl 0701.18007
[14] C. Haase, A. Paffenholz, L.C. Piechnik and F. Santos, Existence of unimodular triangulations – positive results, arXiv:1405.1687.
[15] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics 339, Springer, 1973. · Zbl 0271.14017
[16] M. Kubitzke and V. Welker, Enumerative g-theorems for the Veronese construction for formal power series and graded algebras, Adv. in Appl. Math. 49 (2012), 307-325. · Zbl 1258.13021
[17] M. Leander, Combinatorics of stable polynomials and correlation inequalities, Doctoral Dissertation, University of Stockholm, 2016.
[18] M. Leander, Compatible polynomials and edgewise subdivisions, arXiv:1605.05287.
[19] C.R.F. Maunder, Algebraic Topology, second edition, Cambridge University Press, 1980. · Zbl 0435.55001
[20] T.K. Petersen, Two-sided Eulerian numbers via balls in boxes, Math. Mag. 86 (2013), 159-176. · Zbl 1293.05004
[21] T.K. Petersen, Eulerian Numbers, Birkh¨auser Advanced Texts, Birkh¨auser, 2015.
[22] C.D. Savage and M.J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences, J. Combin. Theory Series A 119 (2012), 850-870. · Zbl 1237.05017
[23] R.P. Stanley, Subdivisions and local h-vectors, J. Amer. Math. Soc. 5 (1992), 805-851. · Zbl 0768.05100
[24] R.P. Stanley, Combinatorics and Commutative Algebra, second edition, Birkh¨auser, Basel, 1996.
[25] P.B. Zhang, On the real-rootedness of the local h-polynomials of edgewise subdivisions of simplices, arXiv:1605.02298. · Zbl 1408.26016
[26] G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, 1995. · Zbl 0823.52002
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