Inequalities for the q-permanent

Bapat, R. B. ; Lal, A. K. (1994) Inequalities for the q-permanent Linear Algebra and its Applications, 197-198 . pp. 397-409. ISSN 0024-3795

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If A is an n×n matrix and q a complex number, then the q-permanent of A is defined as perq A = Σ σ ∈ Sn q1(σ) Πn i = 1 a(i) where Sn 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 perqA ≥ 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 perqA is strictly increasing in [-1, 1]. We propose some more conjectures.

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