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. Roven\c{ t}a and L. E. Temereanca in \cite{ roventa}.
Categories: math.CO math.FA math.RT