2003/04/26 by Stjepan Meljanac, Meljanac, Stjepan, Dragutin Svrtan +1
Mathematics · Physics and Astronomy · #81S05 (47N50 52B30 81R50) #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Random Matrices and Applications #math-ph #math.MP #math.QA #msc:52B30 #msc:81S05
paper · pdf · doi:10.48550/arxiv.math-ph/0304040
63 pages, 1 figure
arxiv created 2003/04/26 · openalex publication_date 2003/04/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study a collections of operators a(k) satisfying the "qkl -canonical commutation relations" a(k)a+(l)-qkla+(l)a(k) =δkl (corresponding for qkl=q to Greenberg (infinite) statistics, for q=± 1 to classical Bose and Fermi statistics).We show that n!× n! matrices An(\qkl\) of scalar products of n-particle states is positive definite for all n if |qkl|<1, all k,l, so that the above commutation relations have a Hilbert space realization. This is achieved by explicit factorizations of An(\qkl\) as a product of matrices of the form (1-QT)± 1, where Q is a diagonal matrix and T is a regular represen- tation of a cyclic matrix. From such factorizations we obtain in Th. 1.9.2 explicit formulas for the determinant of An(\qkl\) in the generic case (which generalizes Zagier's 1-parametric formula). For inversion of An (\qkl\) we use ideas of Božejko and Speicher, and Th.2.2.6 gives a definite answer in terms of maximal chains in subdivision lattices. Our algorithm in Proposition 2.2.18 for computing the entries of An(\qkl\ ) is very efficient. In particular for n=8, when all qkl=q, we found a counterexample to Zagier's conjecture concerning the form of the denominators of the entries in the inverse of An(q). In Cor.2.2.8 we extend Zagier's Conjecture to multiparameter case. By applying a faster algorithm in Prop.2.2.19 we obtain in Th.2.2.20 explicit formulas for the inverse of the matrices An(\qkl\) in the generic case. There are applications of these results to discriminant arrangements of hyperplanes and to contravariant forms of certain quantum groups.