Lih Wang's and Dittert's conjectures on permanents

• Published in: Special Matrices
• Main Authors: Udayan Divya K., Somasundaram Kanagasabapathi
• Format: Article
• Language: English
• Published: De Gruyter 2024-05-01
• Subjects: permanents; doubly stochastic matrices; lih-wang conjecture; phi-maximizing matrix; dittert’s conjecture; 15a15
• Online Access: https://doi.org/10.1515/spma-2024-0006

Let Ωn{\Omega }_{n} denote the set of all doubly stochastic matrices of order nn. Lih and Wang conjectured that for n≥3n\ge 3, per(tJn+(1−t)A)≤t\left(t{J}_{n}+\left(1-t)A)\le tperJn+(1−t){J}_{n}+\left(1-t)perAA, for all A∈ΩnA\in {\Omega }_{n} and all t∈[0.5,1]t\in \left[0.5,1], where Jn{J}_{n} is the n×nn\times n matrix with each entry equal to 1n\frac{1}{n}. This conjecture was proved partially for n≤5n\le 5. Let Kn{K}_{n} denote the set of nonnegative n×nn\times n matrices whose elements have sum nn. Let ϕ\phi be a real valued function defined on Kn{K}_{n} by ϕ(X)=∏i=1nri+∏j=1ncj\phi \left(X)={\prod }_{i=1}^{n}{r}_{i}+{\prod }_{j=1}^{n}{c}_{j} - perXX for X∈KnX\in {K}_{n} with row sum vector (r1,r2,…rn)\left({r}_{1},{r}_{2},\ldots {r}_{n}) and column sum vector (c1,c2,…cn)\left({c}_{1},{c}_{2},\ldots {c}_{n}). A matrix A∈KnA\in {K}_{n} is called a ϕ\phi -maximizing matrix if ϕ(A)≥ϕ(X)\phi \left(A)\ge \phi \left(X) for all X∈KnX\in {K}_{n}. Dittert conjectured that Jn{J}_{n} is the unique ϕ\phi -maximizing matrix on Kn{K}_{n}. Sinkhorn proved the conjecture for n=2n=2 and Hwang proved it for n=3n=3. In this article, we prove the Lih and Wang partially for n=6n=6. It is also proved that if AA is a ϕ\phi -maximizing matrix on K4{K}_{4}, then AA is fully indecomposable.