×

zbMATH — the first resource for mathematics

Worpitzky partitions for root systems and characteristic quasi-polynomials. (English) Zbl 1390.52026
Summary: For a given irreducible root system, we introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the classical formula of Worpitzky.
This partition, and the generalized Eulerian polynomial, recently introduced by Lam and Postnikov, can be used to describe the characteristic (quasi)polynomials of Shi and Linial arrangements. As an application, we prove that the characteristic quasi-polynomial of the Shi arrangement turns out to be a polynomial. We also present several results on the location of zeros of characteristic polynomials, related to a conjecture of Postnikov and Stanley. In particular, we verify the “functional equation” of the characteristic polynomial of the Linial arrangement for any root system, and give partial affirmative results on “Riemann hypothesis” for the root systems of type \(E_6\), \(E_7\), \(E_8\), and \(F_4\).

MSC:
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] C. A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Adv. Math. 122 (1996), no. 2, 193-233. · Zbl 0872.52006
[2] C. A. Athanasiadis, Deformations of Coxeter hyperplane arrangements and their characteristic polynomials. Arrangements–Tokyo 1998, 1-26, Adv. Stud. Pure Math., 27, Kinokuniya, Tokyo, 2000.
[3] C. A. Athanasiadis, Extended Linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley, J. Algebraic Combin. 10 (1999), no. 3, 207-225. · Zbl 0948.52012
[4] C. A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes, Bull. London Math. Soc. 36 (2004), no. 3, 294-302. · Zbl 1068.20038
[5] M. Beck and S. Robins, Computing the continuous discretely. Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics. Springer, New York, 2007. xviii+226 pp. · Zbl 1114.52013
[6] A. Blass and B. Sagan, Characteristic and Ehrhart polynomials, J. Algebraic Combin. 7 (1998), no. 2, 115-126. · Zbl 0899.05003
[7] E. Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol’d]. (French) Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, pp. 21-44. Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973.
[8] L. Comtet, Advanced combinatorics. The art of finite and infinite expansions. Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. xi+343 pp. · Zbl 0283.05001
[9] P. H. Edelman and V. Reiner, Free arrangements and rhombic tilings, Discrete Comput. Geom. 15 (1996), no. 3, 307-340. · Zbl 0853.52013
[10] M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17-76. · Zbl 0803.13010
[11] F. Hirzebruch, Eulerian polynomials, Münster J. Math. 1 (2008), 9-14. · Zbl 1255.11004
[12] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge, 1990. xii+204 pp. · Zbl 0725.20028
[13] H. Kamiya, A. Takemura and H. Terao, Periodicity of hyperplane arrangements with integral coefficients modulo positive integers, J. Algebraic Combin. 27 (2008), no. 3, 317-330. · Zbl 1213.52020
[14] H. Kamiya, A. Takemura and H. Terao, Periodicity of non-central integral arrangements modulo positive integers, Ann. Comb. 15 (2011), no. 3, 449-464. · Zbl 1233.32018
[15] H. Kamiya, A. Takemura and H. Terao, The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements, Arrangements, local systems and singularities, 177-190, Progr. Math., 283, Birkhäuser Verlag, Basel, 2010. · Zbl 1370.32011
[16] T. Lam and A. Postnikov, Alcoved polytopes II. arXiv preprint arXiv:1202.4015 (2012). · Zbl 1134.52019
[17] P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167-189. · Zbl 0432.14016
[18] P. Orlik and H. Terao, Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften, 300. Springer-Verlag, Berlin, 1992. xviii+325 pp. · Zbl 0757.55001
[19] A. Postnikov and R. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 544-597. · Zbl 0962.05004
[20] J. -Y. Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, 1179. Springer-Verlag, Berlin, 1986. x+307 pp. · Zbl 0582.20030
[21] R. Stanley, An introduction to hyperplane arrangements, Geometric combinatorics, 389-496, IAS/Park City Math. Ser., 13, Amer. Math. Soc., Providence, RI, 2007. · Zbl 1136.52009
[22] R. Stanley, Enumerative combinatorics. Volume 1. Second edition, Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 2012. xiv+626 pp.
[23] R. Suter, The number of lattice points in alcoves and the exponents of the finite Weyl groups, Math. Comp. 67 (1998), no. 222, 751-758. · Zbl 0918.20033
[24] H. Terao, Generalized exponents of a free arrangement of hyperplanes andShepherd-Todd-Brieskorn formula, Invent. Math. 63 (1981), no. 1, 159-179. · Zbl 0437.51002
[25] H. Terao, Multiderivations of Coxeter arrangements, Invent. Math. 148 (2002), no. 3, 659-674. · Zbl 1032.52013
[26] J. Worpitzky, Studien über die Bernoullischen und Eulerischen Zahlen, J. Reine Angew. Math. 94 (1883), 203-232.
[27] M. Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math. 157 (2004), no. 2, 449-454. · Zbl 1113.52039
[28] T. Zaslavsky, Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, Memoirs Amer. Math. Soc. 1 (1975), no. 154, vii+102 pp. · Zbl 0296.50010
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.