Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a(k)a(l)=_k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA_n(q) representing the scalar products ofn-particle states is positive definite for alln ifq 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 ofA_n(q) as a product of matrices of the form(1–q^jT)^±1 with 1jn andT a permutation matrix. In particular,A_n(q) is singular if and only ifq^M=1 for some integerM of the formk²–k, 2kn.