Frobenius--Perron dimension and tensor products of algebras

Abstract

In this paper, we study how the Frobenius--Perron dimension of finite-dimensional algebras behaves under tensor products and related constructions. We prove that Frobenius--Perron dimension is super-additive under tensor products and is additive whenever one tensor factor is local. In particular every non-negative integer occurs as a Frobenius--Perron dimension. We further show that the invariant equals $1$ for every representation-infinite cycle-finite algebra, such as a tame concealed or tubular algebra, and we determine it on the grids $\mathsf{k} A_m\otimes_{\mathsf{k}}\mathsf{k} A_n$, where it is $0$, $1$, or $\infty$ according to representation type. Finally we treat skew group algebras of local algebras, for which a McKay quiver computation gives a lower bound and shows that the dimension can jump from finite to infinite.