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 $n\times n$ matrix and let ${\vee }^{k}A$ be its $k$-th symmetric tensor product. We express the normalized trace of ${\vee }^{k}A$ as an integral of the $k$-th powers of the numerical values of $A$ over the unit sphere ${\mathbb{S}}^n$ of ${\mathbb{C}}^n$ with respect to the normalized Euclidean surface measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric polynomials over ${\mathbb{C}}^n$. 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 \cite{cuttler} concerning the monotonicity of products of complete symmetric polynomials. In the process, we give a solution to an open problem that was raised by I. Rovenţa and L. E. Temereanca in \cite{roventa}.