Skip to main content
2 of 5
added related new information about soules!
Suvrit
  • 28.6k
  • 7
  • 82
  • 150

This question is better known as the permanental dominance conjecture and is still an open problem.

According to Zhan's survey, it has been confirmed for every irreducible character of $S_n$ for $n \le 13$. Another reference cited for this conjecture is this survey on open problems about permanents by Cheon and Wanless.


EDIT (added 12/10/2015): Incidentally, the closely related Soules's conjecture whose proof would yield the above permanental dominance (and which states that the largest eigenvalues of the Schur-product matrix of a given Hermitian semidefinite matrix $A$ equals the permanent of $A$), has been very recently shown to be false: check out this explicit counterexample! (please note that the author there incorrectly calls the Soules's conjecture the POT conjecture)

Suvrit
  • 28.6k
  • 7
  • 82
  • 150