Group algebra in characteristic 1 and invariant distances on a finite group

摘要

Invariant metrics on a finite group arise in particular in statistics. They turn out to be closely related to the idempotent elements of the group algebra over the min-plus semifield. The central idempotents (corresponding to bi-invariant metrics) are given by the characters of linear representations of this group. We show that these characters can be obtained from irreducible characters, and more generally, that every idempotent has a unique decomposition as a sum of minimal idempotents. We characterize the minimal idempotents, and construct the irreducible characters from the conjugacy classes of the group. This shows in particular that all the invariant metrics are generated by a finite parametric family of invariant metrics, which are Cayley metrics of cyclic subgroups. The usual distances over are easily recovered from this construction. These result partly carry over to infinite groups.