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

**AMS Subject Classification: **32S22, 17B81,
81R12, 33C70
**Keywords: **arrangement of hyperplanes, period matrix, matroid

debuinus@gmail.com,
anv@email.unc.edu

Annals of Combinatorics 13 (1) pp.115-138 March, 2009

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.
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