A Conjecture on the Permanent of Positive Semidefinite Matrices

Abstract

We state the following conjecture of the author and prove it for n ≤ 5. Let A = (aij) be a complex positive semidefinite n × n matrix with n ≥ 1, and let χλ be an irreducible character of Sn of degree dλ. For σ ∈ Sn, put aσ = ∏i=1n ai,σ(i). The conjecture asserts that ∑σ,τ∈Sn χλ(στ)aσaτ ≤ dλ(per A)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. 2020 Mathematics Subject Classification: 15A15, 15B48, 20C30. Exact proof supplement: https://doi.org/10.5281/zenodo.22976216