Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operators a(k) satisfying the 'q-mutator relations' a(l)a+(k)-qa+(k)a(l)=δk,l (corresponding for q=±1 to classical Bose and Fermi statistics). We show that the n!xn! matrix An(q) representing the scalar products of n-particle states is positive definite for all n if q lies between -1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization of An(q) as a product of matrices of the form (1-qjT)±1 with 1≤j≤n and T a permutation matrix. In particular, An(q) is singular if and only if qM=1 for some integer M of the form k2-k, 2≤k≤n. (orig.)