Inequalities for the q-permanent

Abstract

If A is an n×n matrix and q a complex number, then the q-permanent of A is defined as per_q A = Σ _{σ ∈ Sn} q^1(σ) Πn _{i = 1} a_iσ(i) where S_n is the symmetric group of degree n and l(σ) denotes the number of inversions of σ [i.e., the number of pairs i,j such that 1≤i<j≤n and σ(/)> σ(/)]. The function is of interest in that it includes both the determinant and the permanent as special cases. It is known that if A is positive semidefinite and if -1≤q≤1, then per_qA ≥ 0. We obtain some results for the q-permanent, including Gram's inequality. It has been conjectured by one of the authors that if A is positive definite and not a diagonal matrix, then per_qA is strictly increasing in [-1, 1]. We propose some more conjectures.