Lieb's permanental dominance conjecture

TL;DR

This paper reviews Lieb's permanental dominance conjecture, a long-standing open problem in mathematics, highlighting its significance and recent related advances, but noting it remains unresolved after over fifty years.

Contribution

It provides a comprehensive survey of Lieb's conjecture, its impact, and recent progress on related conjectures, emphasizing the ongoing challenge of proving the original conjecture.

Findings

The conjecture has remained unproven for over 50 years.

Related conjectures have been recently resolved.

The survey summarizes key developments and remaining challenges.

Abstract

We survey the impact of Lieb's influential paper "Proofs of some conjectures on permanents" [J. Math. Mech. 16 1966, 127-134], which introduced the famous permanental dominance conjecture. This conjecture has defied all attacks for over half a century, although a number of related conjectures have recently been resolved.