Yair Lavi

Complex and real counterexamples to an inequality arising from Drury's conjecture

We give counterexamples to the following inequality, stated by Zhang as a consequence of Drury's 2016 permanent conjecture: if A = (ajk)j,k=1n is a Hermitian positive semidefinite matrix of order n ≥ 2, then ∑j=1n−1 |ajn|2 per A[ĵ, n̂] ≤ per A, where A[ĵ, n̂] is the principal sub...

The Kopotun Subpermanent Transform Inequality

We prove the subpermanent transform inequality conjectured by Kopotun in 1996: for every doubly stochastic matrix A of order n ≥ 2 and every 2 ≤ k ≤ n, σk(A) ≥ σk((nJn + A)/(n + 1)). Here σk is the sum of all subpermanents of order k, and Jn has every entry 1/n. Equality holds ex...

A linear gap for the maximal cp-rank

We prove that the maximal cp-rank pₙ satisfies pₙ ≥ n(n−3)/2 for odd n ≥ 5 and pₙ ≥ n(n−3)/2 − 1 for even n ≥ 6. Together with the known upper bound pₙ ≤ n(n+1)/2 − 4, this gives pₙ = n²/2 + O(n). Version 2 (28 September 2026) adds a citation to Hildebrand's order-five copositive...

A linear gap for the maximal cp-rank

We prove that the maximal cp-rank pₙ satisfies pₙ ≥ n(n−3)/2 for odd n ≥ 5 and pₙ ≥ n(n−3)/2 − 1 for even n ≥ 6. Together with the known upper bound pₙ ≤ n(n+1)/2 − 4, this gives pₙ = n²/2 + O(n). Version 2 (28 September 2026) adds a citation to Hildebrand's order-five copositive...

A linear gap for the maximal cp-rank

We prove that the maximal cp-rank pₙ satisfies pₙ ≥ n(n−3)/2 for odd n ≥ 5 and pₙ ≥ n(n−3)/2 − 1 for even n ≥ 6. Together with the known upper bound pₙ ≤ n(n+1)/2 − 4, this gives pₙ = n²/2 + O(n).

A Conjecture on the Permanent of Positive Semidefinite Matrices

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(...