Lieb's permanental dominance conjecture
For a subgroup $G$ of $S_n$ and the character $\chi$ of a nonzero finite-dimensional complex representation of $G$, the normalised generalised matrix function of $A = (a_{ij}) \in \mathbb{C}^{n \times n}$ is $$f_\chi(A) = \frac{1}{\chi(e)} \sum_{\sigma \in G} \chi(\sigma) \prod_{i=1}^n a_{i \sigma(i)},$$ which is real for Hermitian $A$. The determinant, the permanent and the normalised immanants are of this form. The stronger permanent-on-top conjecture is false (Shchesnovich).
Problem. Is $f_\chi(A) \le \operatorname{per}(A)$ for every $n \ge 1$, every Hermitian positive semidefinite $A \in \mathbb{C}^{n \times n}$, every subgroup $G$ of $S_n$ and every such $\chi$?
Source: Lieb (1966), Conjecture α.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- CO, RT
- Keywords
- generalized matrix functions, immanants, OpenProblemsInNLA, permanents
- Literature checked
- Oct. 2, 2026 by clemens-admin
References
@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},
}
@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{shchesnovich2016,
author = {Shchesnovich, Valery S.},
title = {The permanent-on-top conjecture is false},
journal = {Linear Algebra Appl.},
volume = {490},
year = {2016},
pages = {196--201},
doi = {10.1016/j.laa.2015.10.034},
}
@misc{openproblemsinnla-mi11,
author = {Townsend, Alex},
title = {{MI-11}: Lieb's permanental dominance conjecture},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-11/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 It will not change if the problem is solved, reclassified or withdrawn.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-00074, "Lieb's permanental dominance conjecture", clopen.solutions.