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Determinants of the Hypergeometric Period Matrices of an Arrangement and Its Dual
D. Mukherjee and Alexander Varchenko
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
debuinus@gmail.com, anv@email.unc.edu
Annals of Combinatorics 13 (1) pp.115-138 March, 2009
AMS Subject Classification: 32S22, 17B81, 81R12, 33C70
Abstract:
We fix three natural numbers \$k,\, n,\, N\$, such that \$n+k+1=N\$, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of \$N\$ hyperplanes in a \$k\$-dimensional affine space, the other is an arrangement of \$N\$ hyperplanes in an \$n\$-dimensional affine space. We assign weights \$\al_1, \ldots,\, \al_N\$ to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of \$k\$-dimensional hypergeometric integrals and the second is a matrix of \$n\$-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler's gamma function multiplied by a product of exponentials of the weights.
Keywords: arrangement of hyperplanes, period matrix, matroid

References:

1. T.H. Brylawski and J.G. Oxley, The Tutte polynomial and its applications, In: Matroid Applications, Encyclopedia Math. Appl., Vol. 40, Cambridge University Press, Cambridge, (1992) pp. 123-225.

2. A.L. Dixon, On the evaluation of certain denite integrals by means of gamma functions, Proc. London Math. Soc. 3 (1) (1905) 189-205.

3. A. Douai and H. Terao, The determinant of a hypergeometric period matrix, Invent. math. 128 (3) (1997) 417-436.

4. J.G. Oxley, Matroid Theory, Oxford University Press, New York, 1992.

5. T. Muir, A Treatise on the Theory of Determinants, Dover Publications, New York, 1960.

6. E.M. Rains, Transformations of elliptic hypergeometric integrals, Ann. of Math. (2), to appear.

7. A.N. Varchenko, The Euler beta-function, the Vandermonde determinant, Legendre's equation, and critical values of linear functions on a conguration of hyperplanes, I, Math. USSR Izv. 35 (3) (1990) 543-571.

8. A.N. Varchenko, The Euler beta-function, the Vandermonde determinant, Legendre's equation, and critical values of linear functions on a conguration of hyperplanes, II, Math. USSR Izv. 36 (1) (1991) 155-168.

9. T. Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1) (1975) vii+102.