Determinants of Period Matrices and an Application to Selberg's Multidimensional Beta Integral - Richards, Donald; Zheng, Qifu (Advances in applied mathematics, 2002)
In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990), 146–158) defined certain period matrices whose entries are Euler-type integrals representing hypergeometric functions of several variables and derived remarkable closed-form expressions for the determinants of those matrices. In this article, we present elementary proofs of some of Varchenko's determinant formulas. By the same method, we obtain proofs of variations of Varchenko's determinants. As an application, we deduce new proofs of the multidimensional beta integrals of Selberg and of Aomoto. Further, we obtain a new proof of a determinant formula of A. Varchenko (Funct. Anal. Appl.25 (1999), 304–305) in which the entries are multidimensional Selberg-type integrals.