Hanlon and Stanley's Conjecture and the Milnor Fibre of a Braid Arrangement
G. Denham
University of Michigan Department of Mathematics Ann Arbor MI 48109-1109
DOI: 10.1023/A:1008717901994
Abstract
Let A be a real arrangement of hyperplanes. Let B = B(q) be Varchenko's quantum bilinear form of A, introduced [15], specialized so that all hyperplanes have weight q. B(q) is nonsingular for all complex q except certain roots of unity. Here, we examine the kernel of B at roots of unity in relation to the topology of the hyperplane singularity.
We use Varchenko's work [16] to relate B(q) to a Salvetti complex for the Milnor fibration of A. This paper's main result is specific to the arrangement of reflecting hyperplanes associated with the A n - 1 root system. We use a geometric property of the Milnor fibre to resolve a conjecture due to Hanlon and Stanley regarding the \mathfrak S n \mathfrak{S}_n -module structure of the kernel of B(q) at certain roots of unity.
Pages: 227–240
Keywords: hyperplane arrangement; Milnor fibre; quantum bilinear form; braid arrangement
Full Text: PDF
References
1. V.I. Arnol'd, “The cohomology ring of the colored braid group,” Math Notes 5 (1969), 138-140.
2. E. Artal-Bartolo, “Combinatorics and topology of line arrangements in the complex projective plane,” Proc. Amer. Math. Soc. 121(2) (1994), 385-390.
3. D.C. Cohen and A.I. Suciu, “On Milnor fibrations of arrangements,” J. London Math. Soc. (2) 51(1) (1995), 105-119.
4. H. Crapo, “A higher invariant for matroids,” J. Comb. Th. 2 (1967), 406-417.
5. G. Denham and P. Hanlon, “On the Smith normal form of the Varchenko bilinear form of a hyperplane arrangement,” Pacific J. Math. (Special Issue) (1997), 123-146. Olga Taussky-Todd: in memoriam.
6. G. Duchamp, A. Klyachko, D. Krob, and J.-Y. Thibon, “Noncommutative symmetric functions. III. Deformations of Cauchy and convolution algebras,” Discrete Math. Theor. Comput. Sci. 1(1) (1997), 159-216. Lie computations (Marseille, 1994).
7. P. Hanlon and R.P. Stanley, “A q-deformation of a trivial symmetric group action,” Trans. Amer. Math. Soc. 350(11) (1998), 4445-4459.
8. J.W. Milnor, “Infinite cyclic coverings,” in Conference on the Topology of Manifolds, Michigan State Univ., E. Lansing, Mich., 1967, pp. 115-133, Prindle, Weber & Schmidt, Boston, Mass., 1968.
9. J.W. Milnor, Singular Points of Complex Hypersurfaces, Princeton University Press, 1968.
10. M. Newman, Integral Matrices, Academic Press, 1972.
11. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer Verlag,
1992. Grundlehren der Mathematischen Wissenschaften, Vol. 300.
12. R. Randell, “The fundamental group of the complement of a union of complex hyperplanes,” Invent. Math. 69(1) (1982), 103-108.
13. A. Robinson and S. Whitehouse, “The tree representation of σn+1,” J. Pure Appl. Algebra 111(1/3) (1996), 245-253.
14. M. Salvetti, “Topology of the complement of real hyperplanes in Cn ,” Invent. Math. 88 (1987), 603-618.
15. A.N. Varchenko, “Bilinear form of real configuration of hyperplanes,” Adv. Math. 97(1) (1993), 110-144.
16. A.N. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, World Scientific,
1995. Advanced Series in Mathematical Physics, Vol. 21.
17. G.W. Whitehead, Elements of Homotopy Theory, Springer Verlag,
1978. Graduate Texts in Mathematics, Vol. 61.
2. E. Artal-Bartolo, “Combinatorics and topology of line arrangements in the complex projective plane,” Proc. Amer. Math. Soc. 121(2) (1994), 385-390.
3. D.C. Cohen and A.I. Suciu, “On Milnor fibrations of arrangements,” J. London Math. Soc. (2) 51(1) (1995), 105-119.
4. H. Crapo, “A higher invariant for matroids,” J. Comb. Th. 2 (1967), 406-417.
5. G. Denham and P. Hanlon, “On the Smith normal form of the Varchenko bilinear form of a hyperplane arrangement,” Pacific J. Math. (Special Issue) (1997), 123-146. Olga Taussky-Todd: in memoriam.
6. G. Duchamp, A. Klyachko, D. Krob, and J.-Y. Thibon, “Noncommutative symmetric functions. III. Deformations of Cauchy and convolution algebras,” Discrete Math. Theor. Comput. Sci. 1(1) (1997), 159-216. Lie computations (Marseille, 1994).
7. P. Hanlon and R.P. Stanley, “A q-deformation of a trivial symmetric group action,” Trans. Amer. Math. Soc. 350(11) (1998), 4445-4459.
8. J.W. Milnor, “Infinite cyclic coverings,” in Conference on the Topology of Manifolds, Michigan State Univ., E. Lansing, Mich., 1967, pp. 115-133, Prindle, Weber & Schmidt, Boston, Mass., 1968.
9. J.W. Milnor, Singular Points of Complex Hypersurfaces, Princeton University Press, 1968.
10. M. Newman, Integral Matrices, Academic Press, 1972.
11. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer Verlag,
1992. Grundlehren der Mathematischen Wissenschaften, Vol. 300.
12. R. Randell, “The fundamental group of the complement of a union of complex hyperplanes,” Invent. Math. 69(1) (1982), 103-108.
13. A. Robinson and S. Whitehouse, “The tree representation of σn+1,” J. Pure Appl. Algebra 111(1/3) (1996), 245-253.
14. M. Salvetti, “Topology of the complement of real hyperplanes in Cn ,” Invent. Math. 88 (1987), 603-618.
15. A.N. Varchenko, “Bilinear form of real configuration of hyperplanes,” Adv. Math. 97(1) (1993), 110-144.
16. A.N. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, World Scientific,
1995. Advanced Series in Mathematical Physics, Vol. 21.
17. G.W. Whitehead, Elements of Homotopy Theory, Springer Verlag,
1978. Graduate Texts in Mathematics, Vol. 61.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition