×

Extended linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley. (English) Zbl 0948.52012

Summary: A hyperplane arrangement is said to satisfy the “Riemann hypothesis” if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system \(A_{n-1}\). The proof is based on an explicit formula \([1,2,11]\) for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] C.A. Athanasiadis, “Characteristic polynomials of subspace arrangements and finite fields,” Advances in Math.122 (1996), 193-233. · Zbl 0872.52006
[2] C.A. Athanasiadis, “Algebraic combinatorics of graph spectra, subspace arrangements and Tutte polynomials,” Ph.D. Thesis, MIT, 1996.
[3] A. Björner, “Subspace arrangements,” Proc. of the First European Congress of Mathematics, Paris 1992, A. Joseph et al. (Eds.), Progress in Math., Vol. 119, Birkhäuser, 1994, pp. 321-370. · Zbl 0844.52008
[4] A. Björner and T. Ekedahl, “Subspace arrangements over finite fields: cohomological and enumerative aspects,” Advances in Math.129 (1997), 159-187. · Zbl 0896.52021
[5] A. Blass and B.E. Sagan, “Characteristic and Ehrhart polynomials,” J. Alg. Combin.7 (1998), 115-126. · Zbl 0899.05003
[6] H. Crapo and G.-C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, preliminary edition, M.I.T. Press, Cambridge, MA, 1970. · Zbl 0216.02101
[7] Humphreys, J. E., Cambridge Studies in Advanced Mathematics (1990), Cambridge, England · Zbl 0725.20028
[8] M. Jambu and L. Paris, “Combinatorics of inductively factored arrangements,” European J. Combin.16 (1995), 267-292. · Zbl 0823.52012
[9] P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren 300, Springer-Verlag, New York, NY, 1992. · Zbl 0757.55001
[10] A. Postnikov, “Intransitive trees,” J. Combin. Theory Ser. A79 (1997), 360-366. · Zbl 0876.05042
[11] A. Postnikov and R. Stanley, “Deformations of Coxeter Hyperplane Arrangements,” preprint dated April 14, 1997. · Zbl 0962.05004
[12] B.E. Sagan, “Why the Characteristic Polynomial Factors,” Bull. Amer. Math. Soc.36 (1999), 113-133. · Zbl 0921.06001
[13] R. Stanley, “Supersolvable lattices,” Algebra Universalis2 (1972), 197-217. · Zbl 0256.06002
[14] R. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Belmont, CA, 1992.
[15] R. Stanley, “Hyperplane arrangements, interval orders and trees,” Proc. Nat. Acad. Sci.93 (1996), 2620-2625. · Zbl 0848.05005
[16] H. Terao, “Generalized exponents of a free arrangement of hyperplanes and the Shepherd-Todd-Brieskorn formula,” Invent. Math.63 (1981), 159-179. · Zbl 0437.51002
[17] T. Zaslavsky, “Facing up to arrangements: face-count formulas for partitions of space by hyperplanes,” Mem. Amer. Math. Soc.1(154), (1975). · 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. 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.