Remarks on the $\alpha$--permanent

TL;DR

This paper explores properties of the $oldsymbol{ ext{α}}$--permanent, establishing new expansion formulas, inequalities, and partial verifications of conjectures related to its nonnegativity for positive semi-definite matrices.

Contribution

It introduces new expansion formulas for the $oldsymbol{ ext{α}}$--permanent, proves Lieb-type inequalities, and verifies Shirai's nonnegativity conjecture up to 5x5 matrices.

Findings

Established expansion formulas for the α-permanent.

Proved Lieb-type inequalities for specific matrix classes.

Verified Shirai's nonnegativity conjecture up to 5x5 matrices.

Abstract

We recall Vere-Jones's definition of the $\alpha$--permanent and describe the connection between the (1/2)--permanent and the hafnian. We establish expansion formulae for the $\alpha$--permanent in terms of partitions of the index set, and we use these to prove Lieb-type inequalities for the $\pm\alpha$--permanent of a positive semi-definite Hermitian $n\times n$ matrix and the $\alpha/2$--permanent of a positive semi-definite real symmetric $n\times n$ matrix if $\alpha$ is a nonnegative integer or $\alpha\ge n-1$. We are unable to settle Shirai's nonnegativity conjecture for $\alpha$--permanents when $\alpha\ge 1$, but we verify it up to the $5\times 5$ case, in addition to recovering and refining some of Shirai's partial results by purely combinatorial proofs.