Lattice theory and toeplitz determinants

Abstract

This is a survey of our recent joint investigations of lattices that are generated by finite Abelian groups. In the case of cyclic groups, the volume of a fundamental domain of such a lattice is a perturbed Toeplitz determinant with a simple Fisher–Hartwig symbol. For general groups, the situation is more complicated, but it can still be tackled by pure matrix theory. Our main result on the lattices under consideration states that they always have a basis of minimal vectors, while our results in the other direction concern exact and asymptotic formulas for perturbed Toeplitz determinants. The survey is a slightly modified version of the talk given by the first author at the Humboldt Kolleg and the IWOTA in Tbilisi in 2015. It is mainly for operator theorists and therefore also contains an introduction to the basics of lattice theory.