Generalization of some integrals over unitary matrices by character expansion of groups

Abstract

The character expansion method was introduced by Balantekin [Phys. Rev. D 62, 085017 (2000)] for integration over the unitary group and, in particular, for calculating the well-known Harish–Chandra–Itzykson–Zuber integral where the coefficient matrices in the integrand are square matrices with nonzero determinants. However, in some applications such as the capacity analysis of multiple-input multiple-output channels in wireless communications and information theory, or applying the color-flavor transformation to lattice quantum chromodynamics in physics, or the theory of random matrices in mathematics, the integration over the unitary group is required where general rectangular complex matrices appear in the integrand. In this paper, we use the character expansion of groups to generalize two integrals over the unitary group that have general rectangular complex matrices in the integrand. Although we consider only two integrals, we believe that the integration framework presented here can be used for other integrals over unitary matrices.