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

TL;DR

This paper derives an integral formula for the normalized trace of symmetric tensor powers of matrices, connecting it to symmetric polynomials, and applies it to prove a combinatorial theorem and address an open problem in polynomial monotonicity.

Contribution

It introduces a novel integral representation for the trace of symmetric tensor powers, enabling new proofs and generalizations in combinatorics and polynomial analysis.

Findings

Provided an integral expression for the normalized trace of symmetric tensor powers.

Presented a new proof of the MacMahon Master Theorem.

Solved an open problem on the monotonicity of products of complete symmetric polynomials.

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. Roven\c{t}a and L. E. Temereanca in \cite{roventa}.