Hook immanantal inequalities for Hadamard's function
Let \(A=[a_{ij}]\) be a positive semi-definite matrix of order \(n\). The Hadamard function is \(h(A)=\prod_{i=1}^n a_{ii}\). The immanants are a family of matrix functions including the permanent and determinant. Each immanant on matrices of order \(n\) is associated with an irreducible character of the symmetric group \(S_n\) and hence with a partition of \(n\). The normalized hook immanant \({\overline d}_k\) is associated with the hook partition \((k,1^{n-k})\). \textit{P. Heyfron} [Linear Multilinear Algebra 24, No. 1, 65-78 (1988; Zbl 0678.15009)] showed that \[ \text{ per}(A)={\overline d}_n(A) \geq {\overline d}_{n-1}(A) \geq \cdots \geq {\overline d}_2(A) \geq {\overline d}_1(A) = \text{det}(A). \] This paper seeks to find where the Hadamard function lies in this descending sequence in the case when \(A\) is the Laplacian matrix of a tree with \(n\) vertices. Specificially, it seeks \(\kappa(A)\) such that \({\overline d}_{\kappa(A)}(A) \geq h(A) > {\overline d}_{\kappa(A)-1}(A)\). It is shown that \(\lceil n/2+m/3 \rceil \geq \kappa(A) \geq \lceil(n+1)/2\rceil\) where \(m\) is the size of the largest matching in the tree. Both bounds are achieved, and it is shown that the term \(m/3\) in the upper bound cannot be replaced by \(m/4\) asymptotically.
scientific article; zbMATH DE number 1399451