Marcus's inequality for the permanent of block permanents
Let $\operatorname{per}$ be the permanent. For integers $m, k \ge 2$, partition $A \in \mathbb{C}^{mk \times mk}$ into an $m \times m$ array of $k \times k$ blocks $A_{ij}$, and let $G$ be the $m \times m$ matrix with entries $g_{ij} = \operatorname{per}(A_{ij})$. Lieb proved the inequality below for $m = 2$, and showed that it would follow from his permanental dominance conjecture. For real $A$, Pate showed that for each $m$ it holds for all but finitely many $k$.
Problem. Is $\operatorname{per}(A) \ge \operatorname{per}(G)$ for all integers $m, k \ge 2$ and every Hermitian positive semidefinite $A \in \mathbb{C}^{mk \times mk}$?
Source: Marcus, in Marcus and Minc (1965), Conjecture 9.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- CO
- Keywords
- block matrices, OpenProblemsInNLA, permanents
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{marcus1965,
author = {Marcus, Marvin and Minc, Henryk},
title = {Permanents},
journal = {Amer. Math. Monthly},
volume = {72},
number = {6},
year = {1965},
pages = {577--591},
doi = {10.1080/00029890.1965.11970575},
}
@article{lieb1966,
author = {Lieb, Elliott H.},
title = {Proofs of some conjectures on permanents},
journal = {J. Math. Mech.},
volume = {16},
number = {2},
year = {1966},
pages = {127--134},
doi = {10.1512/iumj.1967.16.16008},
}
@article{pate1982,
author = {Pate, Thomas H.},
title = {Inequalities relating groups of diagonal products in a {G}ram matrix},
journal = {Linear Multilinear Algebra},
volume = {11},
number = {1},
year = {1982},
pages = {1--17},
doi = {10.1080/03081088208817427},
}
@incollection{wanless2022,
author = {Wanless, Ian M.},
title = {Lieb's permanental dominance conjecture},
booktitle = {The Physics and Mathematics of {E}lliott {L}ieb, Vol. {II}},
editor = {Frank, Rupert L. and Laptev, Ari and Lewin, Mathieu and Seiringer, Robert},
publisher = {EMS Press},
year = {2022},
pages = {501--516},
doi = {10.4171/90-2/48},
eprint = {2202.01867},
archivePrefix = {arXiv},
}
@article{zhang2016,
author = {Zhang, Fuzhen},
title = {An update on a few permanent conjectures},
journal = {Spec. Matrices},
volume = {4},
number = {1},
year = {2016},
pages = {305--316},
doi = {10.1515/spma-2016-0030},
eprint = {1608.02844},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi12,
author = {Townsend, Alex},
title = {{MI-12}: Marcus's inequality for the permanent of block permanents},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-12/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00141, "Marcus's inequality for the permanent of block permanents", clopen.solutions.