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arXiv · 2106.01486

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

Hassan Issa·Hassan Abbas·Bassam Mourad
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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}.

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Hassan Issa, Hassan Abbas, Bassam Mourad. 2021-06-02. On an integral representation of the normalized trace of the $k$-th symmetric tensor power of matrices and some applications. https://arxiv.org/abs/2106.01486

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