Study of Gram Matrices in Fock Representation of Multiparametric Canonical Commutation Relations, Extended Zagier's Conjecture, Hyperplane Arrangements and Quantum Groups

TL;DR

This paper thoroughly investigates Gram matrices in Fock representations of multiparametric canonical commutation relations, extending Zagier's conjecture, solving the inversion problem, and providing a counterexample to the original conjecture.

Contribution

It extends Zagier's formulas to multiparametric cases, solves the Gram matrix inversion problem, and presents a counterexample to Zagier's original conjecture.

Findings

Extended Zagier's formulas for multiparametric systems

Complete solution to the Gram matrix inversion problem

Counterexample to the original Zagier's conjecture for n=8

Abstract

In this Colloqium Lecture (by one of the authors (D.S)) a thorough presentation of the authors' research on the subjects, stated in the title, is given. By quite laborious mathematics it is explained how one can handle systems in which each Heisenberg commutation relation is deformed separately. For Hilbert space realizability a detailed determinant computations (extending Zagier's one-parameter formulas) are carried out. The inversion problem of the associated Gram matrices on Fock weight spaces is completely solved (Extended Zagier's conjecture) and a counterexample (for $n=8$) to the original Zagier's conjecture is presented in detail.