Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley
Christos A. Athanasiadis
DOI: 10.1023/A:1018778031648
Abstract
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.
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.Pages: 207–225
Keywords: hyperplane arrangement; characteristic polynomial; root system
Full Text: PDF
References
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2. C.A. Athanasiadis, “Algebraic combinatorics of graph spectra, subspace arrangements and Tutte polynomials,” Ph.D. Thesis, MIT, 1996.
3. A. Bj\ddot orner, “Subspace arrangements,” Proc. of the First European Congress of Mathematics, Paris 1992, A. Joseph et al. (Eds.), Progress in Math., Vol. 119, Birkh\ddot auser, 1994, pp. 321-370.
4. A. Bj\ddot orner and T. Ekedahl, “Subspace arrangements over finite fields: cohomological and enumerative aspects,” Advances in Math. 129 (1997), 159-187.
5. A. Blass and B.E. Sagan, “Characteristic and Ehrhart polynomials,” J. Alg. Combin. 7 (1998), 115-126.
6. H. Crapo and G.-C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, preliminary edition, M.I.T. Press, Cambridge, MA, 1970.
7. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, England, 1990.
8. M. Jambu and L. Paris, “Combinatorics of inductively factored arrangements,” European J. Combin. 16 (1995), 267-292.
9. P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren 300, Springer-Verlag, New York, NY, 1992.
10. A. Postnikov, “Intransitive trees,” J. Combin. Theory Ser. A 79 (1997), 360-366.
11. A. Postnikov and R. Stanley, “Deformations of Coxeter Hyperplane Arrangements,” preprint dated April 14, 1997.
12. B.E. Sagan, “Why the Characteristic Polynomial Factors,” Bull. Amer. Math. Soc. 36 (1999), 113-133.
13. R. Stanley, “Supersolvable lattices,” Algebra Universalis 2 (1972), 197-217.
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.
16. H. Terao, “Generalized exponents of a free arrangement of hyperplanes and the Shepherd-Todd-Brieskorn formula,” Invent. Math. 63 (1981), 159-179.
17. T. Zaslavsky, “Facing up to arrangements: face-count formulas for partitions of space by hyperplanes,” Mem. Amer. Math. Soc. 1(154), (1975).