Determinants and Inversion of Gram Matrices in Fock Representation of $\{q_{kl}\}$- Canonical Commutation Relations and Applications to Hyperplane Arrangements and Quantum Groups. Proof of an Extension of Zagier's Conjecture

Abstract

In this paper we study a collections of operators $a(k)$ satisfying the "$q_{kl} $-canonical commutation relations" $a(k)a^{+}(l)-q_{kl}a^{+}(l)a(k) =\delta_{kl} $ (corresponding for $q_{kl}=q$ to Greenberg (infinite) statistics, for $q=\pm 1$ to classical Bose and Fermi statistics).We show that $n!\times n!$ matrices $A_{n}(\{q_{kl}\})$ of scalar products of n-particle states is positive definite for all n if $|q_{kl}|<1$, all k,l, so that the above commutation relations have a Hilbert space realization. This is achieved by explicit factorizations of $A_{n}(\{q_{kl}\})$ as a product of matrices of the form $(1-QT)^{\pm 1}$, 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 $A_{n}(\{q_{kl}\})$ in the generic case (which generalizes Zagier's 1-parametric formula). For inversion of $A_{n} (\{q_{kl}\})$ we use ideas of Bo\v{z}ejko 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 $A_{n}(\{q_{kl}\} )$ is very efficient. In particular for $n=8$, when all $q_{kl}=q$, we found a counterexample to Zagier's conjecture concerning the form of the denominators of the entries in the inverse of $A_{n}(q)$. 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 $A_n(\{q_{kl}\})$ in the generic case. There are applications of these results to discriminant arrangements of hyperplanes and to contravariant forms of certain quantum groups.