The norm of the symmetric product of tensors and its applications to permanents

Abstract

Let $A$ be an $np \times np$ positive semidefinite matrix partitioned into $p \times p$ blocks $A_{ij}$, and $P$ be the $n \times n$ matrix with $p_{ij} = {\rm per}(A_{ij})$. Marcus conjectures in 1965 that ${\rm per}(A) \geq {\rm per}(P)$, which remains open in general. In this paper, we prove a generalized Neuberger identity, which provides an expression for the norm of the symmetric product of tensors. The proof involves a careful analysis of double cosets of symmetric groups with respect to Young subgroups. Then, we leverage it to the Marcus's conjecture for block tridiagonal matrices.