Determinants, permanents and some applications to statistical shape theory (Q1931848)

Let \(|\mathbf{A}|\) and \([\mathbf{A}]\) denote the determinant and, respectively, the permanent of a square matrix \(\mathbf{A}=(a_{ij})\). From the introduction (with minor changes): Modern computing technology can use existing definitions and algorithms to compute \(|\mathbf{A}|\) and \([\mathbf{A}]\) quickly and efficiently. However, from a theoretical viewpoint, it is of interest to propose representations for \(|\mathbf{A}|\) and \([\mathbf{A}]\) in terms of~\(\mathbf{A}\), instead of the usual representation in terms of the elements~\(a_{ij}\) or submatrices. In this paper, some solutions are provided to these problems. In Section~2, an expansion for~\(|\mathbf{A}|\) as a function of powers of traces, and a formula for integer powers of~\(|\mathbf{A}|\) are presented. They are simplified for positive definite matrices, and are expanded in terms of zonal polynomials. Section~3 uses these formulas to establish a connection between \(|\mathbf{A}|\) and~\([\mathbf{A}]\). Section~4 applies the formulas for powers of~\(|\mathbf{A}|\) to random matrix theory based on elliptical distributions with kernels that depend on determinants. Some integrals, similar to those given in case of traces by \textit{C.~S.~Herz} [Ann. Math. (2) 61, 474--523 (1955; Zbl 0066.32002)] are also derived. Finally, an application to statistical shape theory is discussed by deriving the central nonisotropic configuration density in terms of zonal polynomials.