On tensor products of path algebras of type A

We derive a formula for the Coxeter polynomial of the s-fold tensor product F[A_{n_1-1}] x ... x F[A_{n_s-1}] of path algebras of linearly oriented quivers of Dynkin type A_{n_i-1}, in terms of the weights n_1, ..., n_s > 1, and show that conversely the weights can be recovered from the Coxeter polynomial of the tensor product. Our results have applications in singularity theory, in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles.