On the normal forms of modules with respect to parametrizing bimodules

We present a general solution of the isomorphism and multiplicity problems, restricted to the class of all modules lying in homogeneous tubes, for tame algebras (Theorem 2.4). We introduce a notion of the characteristic polynomial of a module, which plays an analogous role as in the classical situation. This notion uses a Smith form Δ(HB(M))Δ(HB(M)) of certain polynomial matrix HB(M)HB(M) associated to a module M and a bimodule B   parametrizing a family of homogeneous modules FF. We show that Δ(HB(M))Δ(HB(M)) encodes an essential information on a decomposition of M   into a direct sum of indecomposables from FF ( Theorem 2.2 and Theorem 2.3).