Complex and real counterexamples to an inequality arising from Drury’s conjecture

2026-10-19

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Research publication

Complex and real counterexamples to an inequality arising from Drury’s conjecture

Preprint

Original date

October 19, 2026

Authors

Yair Lavi

Topics

• Matrix theory
• Permanents
• Positive semidefinite matrices
• Matrix inequalities

External publication links

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Abstract

We give counterexamples to the following inequality, stated by Zhang as a consequence of Drury’s 2016 permanent conjecture: if \(A=(a_{jk})_{j,k=1}^n\) is a Hermitian positive semidefinite matrix of order \(n\geq2\), then \[ \sum_{j=1}^{n-1}|a_{jn}|^2\operatorname{per}A[\,\hat{\jmath},\hat{n}\,]\leq\operatorname{per}A \] where \(A[\,\hat{\jmath},\hat{n}\,]\) is the principal submatrix obtained by deleting indices \(j\) and \(n\). Our complex example has order twelve and rank two; our real example has order twenty-one and rank three.

Related paper

Section 6 relates the inequality to my conjecture in A Conjecture on the Permanent of Positive Semidefinite Matrices. It proves that if the inequality holds for each index, then so does the standard irrep case of my conjecture, which is equivalent to: \[ \sum_{i,j=1}^{n}|a_{ij}|^2\,|\operatorname{per}A(i|j)|^2\leq n\,(\operatorname{per}A)^2, \] where \(A(i|j)\) is the submatrix obtained by deleting row \(i\) and column \(j\). Both counterexamples violate the inequality but do not disprove my conjecture for the standard representation.

Paper

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