Hook immanantal inequalities for Hadamard's function
type: Item
instance of: scholarly article
title: Hook immanantal inequalities for Hadamard's function
zbMATH Open document ID: 0948.15010
DOI: 10.1016/S0024-3795(99)00180-9
author: Onn Chan
author: Bee-San Ng
published in: Linear Algebra and its Applications
publication date: 2000-08-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 1399451
zbMATH Keywords: immanant inequality
zbMATH Keywords: Hadamard's function
zbMATH Keywords: Laplacian of trees
zbMATH Keywords: permanent
zbMATH Keywords: determinant
zbMATH Keywords: hook immanant
reviewed by: Ian M. Wanless
MaRDI profile type: Publication
Wikidata QID: Q127968442
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description: scientific article; zbMATH DE number 1399451