On an integral representation of the normalized trace of the k-th symmetric tensor power of matrices and some applications

Abstract

Let A be an matrix and let be its k-th symmetric tensor power. We express the normalized trace of as an integral of the k-th powers of the numerical values of A over the unit sphere of with respect to the rotation-invariant probability measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric polynomials over . As applications, we present a new proof for the MacMahon Master Theorem in enumerative combinatorics. Then, our next application deals with a generalization of the work of Cuttler et al. in [Cuttler A, Greene C, Skandera M. Inequalities for symmetric means. Eur J Comb. 2011;32(6):745–761] concerning the monotonicity of products of complete symmetric polynomials. Finally, we give a solution to an open problem that was raised by Rovena and Temereanca in [Roventa I, Temereanca LE. A note on the positivity of the even degree complete homogeneous symmetric polynomials. Mediterr J Math. 2019;16(1):1–16].