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 power. 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 rotation-invariant probability 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 [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].

Keywords:

• Integral formula
• trace of symmetric powers
• homogeneous polynomials

Acknowledgements

We are grateful to the reviewer for his/her valuable suggestions and comments. We highly appreciate his/her careful corrections on the original version of the paper. The first author would like to thank Prof. Dr W. Bauer for his stress on the importance of Lemma 2.3 during several conversations. The first author would also like to dedicate his work to his family and to Ibn L. Hassan Al-Moaammal.

Disclosure statement

No potential conflict of interest was reported by the author(s).