INEQUALITIES FOR THE Q-PERMANENT

If A is an n X n matrix and q a complex number, then the q-permanent o f A is defined as [GRAPHICS] where S(n) is the symmetric group of degr ee n and l(sigma) denotes the number of inversions of sigma [i.e., the number of pairs i, j such that 1 less-than-or-equal-to i < j less-tha n-or-equal-to n and sigma(i) > a (j)]. The function is of interest in that it includes both the determinant and the permanent as special cas es. It is known that if A is positive semidefinite and if -1 less-than -or-equal-to q less-than-or-equal-to 1, then per(q) A greater-than-or- equal-to 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(q) A is stri ctly increasing in [-1, 1]. We propose some more conjectures.