An inequality involving permanents of certain direct products
Let \({\mathbb{C}}\) be the complex field and \(T_ n\) the collection of all n- linear complex valued functions on \({\mathbb{C}}^ m\). Let \(S_ n\) denote the subspace of \(T_ n\) consisting of all fully symmetric members of \(T_ n\). The usual inner product of \({\mathbb{C}}^ m\) can be extended in the standard way to \(T_ n\). Let \({\mathcal S}_ n\) be the symmetric group acting on \(\{\) 1,...,\(n\}\). If \(A\in T_ n\), \(\sigma\in {\mathcal S}_ n\), then, by definition \(A_{\sigma}(z_ 1,...,z_ n)=A(z_{\sigma(1)},...,z_{\sigma(n)})\) and \[ P(A)=\frac{1}{n!}\sum_{\sigma \in {\mathcal S}_ n}A_{\sigma}. \] If \(A\in T_ n\) and \(B\in T_ p\), A. B is defined to be \(P(A\otimes B)\). The main result states that if A is a decomposable member of \(S_ n\), then \[ \| A.A\|^ 2\geq 2^ n\left( \begin{matrix} 2n\\ n\end{matrix} \right)^{-1}\| A\otimes A\|^ 2. \] As a consequence an inequality for permanents is obtained.
scientific article; zbMATH DE number 3843927