HEXAGONbeta

A Conjecture on the Permanent of Positive Semidefinite Matrices

A mix of human-written and AI-generated textHuman understanding: all partsmath.CO — Combinatorics

Contributed by Yair Lavi ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

We state the following conjecture of the author and prove it for n≤5n\leq5. Let A=(aij)A=(a_{ij}) be a complex positive semidefinite n×nn\times n matrix with n≥1n\geq1, and let χλ\chi_\lambda be an irreducible character of SnS_n of degree dλd_\lambda. For σ∈Sn\sigma\in S_n, put aσ=∏i=1nai,σ(i)a_\sigma=\prod_{i=1}^n a_{i,\sigma(i)}. The conjecture asserts that ∑σ,τ∈Snχλ(στ)aσaτ≤dλ(per⁡A)2.\sum_{\sigma,\tau\in S_n}\chi_\lambda(\sigma\tau)a_\sigma a_\tau \leq d_\lambda(\operatorname{per}A)^2. We show that Soules' permanent-on-top conjecture implies this inequality, and that it in turn implies both Lieb's immanant dominance and Chollet's permanent inequality. We also settle the previously unresolved order-four case of Soules' conjecture.

Author comments: The supplement cited in the text (exact certificates, order-four proof inputs and checking programs; ZIP, 176.5 MB) is in the Zenodo record https://doi.org/10.5281/zenodo.22976216.

Tools used

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

References

  1. E. H. Lieb, ``Proofs of Some Conjectures on Permanents,'' Journal of Mathematics and Mechanics 16 (1966), 127--134. doi:10.1512/iumj.1967.16.16008 (https://doi.org/10.1512/iumj.1967.16.16008).DOI
  2. R. B. Bapat and V. S. Sunder, ``An Extremal Property of the Permanent and the Determinant,'' Linear Algebra and its Applications 76 (1986), 153--163. doi:10.1016/0024-3795(86)90220-X (https://doi.org/10.1016/0024-3795(86)90220-X).DOI
  3. G. W. Soules, ``An Approach to the Permanental-Dominance Conjecture,'' Linear Algebra and its Applications 201 (1994), 211--229. doi:10.1016/0024-3795(94)90117-1 (https://doi.org/10.1016/0024-3795(94)90117-1).DOI
  4. V. S. Shchesnovich, ``The Permanent-on-Top Conjecture Is False,'' Linear Algebra and its Applications 490 (2016), 196--201. doi:10.1016/j.laa.2015.10.034 (https://doi.org/10.1016/j.laa.2015.10.034).DOI
  5. J. Chollet, ``Is There a Permanental Analogue to Oppenheim's Inequality?,'' American Mathematical Monthly 89 (1982), 57--58. doi:10.1080/00029890.1982.11995380 (https://doi.org/10.1080/00029890.1982.11995380).DOI
  6. M. Marcus, ``The Permanent Analogue of the Hadamard Determinant Theorem,'' Bulletin of the American Mathematical Society 69 (1963), 494--496. doi:10.1090/S0002-9904-1963-10975-1 (https://doi.org/10.1090/S0002-9904-1963-10975-1).DOI
  7. T. H. Pate, ``Immanant Inequalities and Partition Node Diagrams,'' Journal of the London Mathematical Society (2) 46 (1992), 65--80. doi:10.1112/jlms/s2-46.1.65 (https://doi.org/10.1112/jlms/s2-46.1.65).DOI
  8. I. M. Wanless, ``Lieb's Permanental Dominance Conjecture,'' in The Physics and Mathematics of Elliott Lieb , vol. 2, R. L. Frank, A. Laptev, M. Lewin, and R. Seiringer, eds., EMS Press, Berlin, 2022, 501--516. doi:10.4171/90-2/48 (https://doi.org/10.4171/90-2/48).DOI
  9. R. Grone and S. Pierce, ``Permanental Inequalities for Correlation Matrices,'' SIAM Journal on Matrix Analysis and Applications 9 (1988), 194--201. doi:10.1137/0609016 (https://doi.org/10.1137/0609016).DOI
  10. G. W. Soules, Matrix Functions and the Laplace Expansion Theorem , Ph.D. dissertation, University of California, Santa Barbara, 1966.
Show all 15 referencesShow fewer references
  1. T. H. Pate, ``Row Appending Maps, Ψ\Psi-Functions, and Immanant Inequalities for Hermitian Positive Semi-Definite Matrices,'' Proceedings of the London Mathematical Society (3) 76 (1998), 307--358. doi:10.1112/S0024611598000100 (https://doi.org/10.1112/S0024611598000100).DOI
  2. G. Hutchinson, ``An Elementary Proof of Chollet's Permanent Conjecture for 4×44\times4 Real Matrices,'' Special Matrices 9 (2021), 83--102. doi:10.1515/spma-2020-0126 (https://doi.org/10.1515/spma-2020-0126).DOI
  3. K. Rodtes, ``Chollet's Permanent Conjecture for 4×44\times4 Matrices,'' Linear and Multilinear Algebra 72 (2024), no. 16, 2633--2638. doi:10.1080/03081087.2023.2279150 (https://doi.org/10.1080/03081087.2023.2279150).DOI
  4. Q. Li, Q. Zhao, and W. Luo, ``Chollet's Permanent Conjecture Through Order Six via Border Induction,'' Preprints.org 202608.2196/v1 (31 August 2026), unrefereed preprint. doi:10.20944/preprints202608.2196.v1 (https://doi.org/10.20944/preprints202608.2196.v1).DOI
  5. Y. Li, ``Lieb's Permanental Dominance Conjecture for Ordinary Immanants through Order Fifteen,'' arXiv:2609.13412 (2026), preprint. arXiv:2609.13412 (https://arxiv.org/abs/2609.13412).arXiv

Version history

  1. v1 · 8 pagesSubmitted by Yair LaviThe supplement cited in the text (exact certificates, order-four proof inputs and checking programs; ZIP, 176.5 MB) is in the Zenodo record https://doi.org/10.5281/zenodo.22976216.CurrentNov 13, 2026