Torsion Units of Integral Group Rings and Scheme Rings

Abstract

We study torsion units of algebras over the ring of integers Z with nice bases. These include integral group rings, integral adjacency algebras of association schemes and integral C-algebras. Torsion units of group rings have been studied extensively since the 1960’s. Much of the attention has been devoted to the Zassenhaus conjecture for normalized torsion units of ZG, which says that they should be rationally conjugate (i.e. in QG) to elements of the groupG. In recent years several new restrictions on integral partial augmentations for torsion units of ZG have been introduced that have improved the e ectiveness of the Luthar-Passi method for checking the Zassenhaus conjecture for specific finite groups G. We have implemented a computer program that constructs units of QG that have integral partial augmentations that are relevant to the Zassenhaus conjecture. Indeed, any unit of ZG with these partial augmentations would be a counterexample to the conjecture. In all but three exceptions among groups of order less than 160, we have constructed units of QG with these partial augmentations that satisfy a condition which implies they cannot be rationally conjugate to an element of ZG. Currently our package has computational di culties with the Luthar-Passi method for some of the groups of order 160. As C-algebras are generalization of groups, it is natural to ask about torsion units of C-algebras. We establish some basic results about torsion units of Calgebras analogous to what happens for torsion units of group rings. These results can be immediately applied to give new results for Schur rings, Hecke algebras, adjacency algebras of association schemes and fusion rings. We also investigate the possibility for a conjecture analogous to the Zassenhaus conjecture in the Calgebra setting.