On Frobenius-Perron dimension

Abstract

We propose a notion of Frobenius-Perron dimension for certain free Z \mathbb {Z} -modules of infinite rank and compute it for the Z \mathbb {Z} -modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. The definition of Frobenius-Perron dimension that we are introducing naturally generalizes the well-known Frobenius-Perron dimension on the category of finite dimensional complex representations of a finite group.