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 × n matrix and let ∨ k A be its k-th symmetric tensor product. We express the normalized trace of ∨ k A as an integral of the k-th powers of the numerical values of A over the unit sphere S n of 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 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 [10] 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. Rovent ¸a and L. E. Temereanca in [31].

New applications to combinatorics and invariant matrix norms of an integral representation of natural powers of the numerical values

“…Integrating both sides of the preceding equality w.r.t. µ over S n , and using Lemma 2.1 in [23] we get…”

“…(2.9) Motivated by the above result and similar to Example 2.1 in [23], we express N k in terms of Schatten norms for the cases k = 3, 4 and 5 (the cases k = 1 or k = 2 can be obtained directly from the results in [10]).…”

“…. In [23], we have expressed T r(∨ k A) as an integral of the k-th powers of the numerical values of A over the unit sphere S n of C n with respect to the normalized Euclidean surface measure σ. More precisely, we have proved the following.…”

“…In this paper, we continue to explore the potential consequences of (1.1) in many directions. For convenience, we shall use the same terminology and notation as in [23] and we shall always assume that k is non-zero.…”

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Let ∨ k A be the k-th symmetric tensor power of A ∈ Mn(C). In [23], we have expressed the normalized trace of ∨ k A as an integral of the k-th powers of the numerical values of A over the unit sphere S n of C n with respect to the normalized Euclidean surface measure σ. In this paper, we first use this integral representation to construct a family of unitarily invariant norms on Mn(C) and then explore their relations to Schatten-norms of ∨ k A.

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