A Conjecture on the Permanent of Positive Semidefinite Matrices
2026-09-26
Research publication
A Conjecture on the Permanent of Positive Semidefinite Matrices
Preprint
- Original date
- Authors
- Yair Lavi
- Combinatorics
- Permanents
- Matrix inequalities
About this work
I’m excited to share a conjecture I’ve been waiting five years to publish!
We state the following conjecture of mine and prove it for \(n\leq5\). Let \(A=(a_{ij})\) be a complex positive semidefinite \(n\times n\) matrix with \(n\geq1\), and let \(\chi_\lambda\) be an irreducible character of \(S_n\) of degree \(d_\lambda\). For \(\sigma\in S_n\), put \(a_\sigma=\prod_{i=1}^n a_{i,\sigma(i)}\). The conjecture asserts that
\[ \sum_{\sigma,\tau\in S_n}\chi_\lambda(\sigma\tau)a_\sigma a_\tau \leq d_\lambda\bigl(\operatorname{per}A\bigr)^2. \]
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.
Supplementary material
The exact certificates, the complete order-four proof inputs, and the programs that check them are archived on Zenodo as doi:10.5281/zenodo.22976216.
SHA-256 of the ZIP: 8e85c86caeac2b99d001789763d9ee63d5bcaf47ff469f730a4d97e65cae2947