HEXAGONbeta

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

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatoricsmath.RA — Rings and Algebras

Contributed by Yair Lavi ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

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

Tools used

OpenAI
GPT-5.6-sol
OpenAI
GPT-6 Astra
Anthropic
Claude FableVersion 5
Anthropic
Claude FableVersion 5.1

References

  1. F. Zhang, An update on a few permanent conjectures , Special Matrices 4 (1) (2016), 305--316. doi:10.1515/spma-2016-0030 (https://doi.org/10.1515/spma-2016-0030). See also arXiv:1608.02844v1, p. 14, equation (21).arXivDOI
  2. G. Hutchinson, A counterexample to the Drury permanent conjecture , Special Matrices 5 (1) (2017), 301--302. doi:10.1515/spma-2017-0022 (https://doi.org/10.1515/spma-2017-0022).DOI
  3. L. R. Chalmers, A real counterexample to the Drury permanent conjecture , preprint, version 2, 13 August 2026. doi:10.5281/zenodo.21910577 (https://doi.org/10.5281/zenodo.21910577).DOI
  4. Y. Lavi, A conjecture on the permanent of positive semidefinite matrices , Zenodo preprint (2026). doi:10.5281/zenodo.23001182 (https://doi.org/10.5281/zenodo.23001182).DOI

Version history

  1. v1 · 10 pagesSubmitted by Yair LaviInitial depositCurrentNov 07, 2026