Decomposition Classes for Representations of Tame Quivers

For the variety of representations of a tame quiver, we describe a finite stratification into smooth subvarieties that are invariant under the group of base changes and that admit smooth rational geometric quotients. A stratum consists of all representations that differ only in the parameters of the homogeneous indecomposable direct summands. For the special case of the loop our definition coincides with the well-known bundles (V. I. Arnold, Uspekhi Mat. Nauk26 (1971), 101–114) or decomposition classes of matrices (W. Borho and H. Kraft, Comment. Math. Helv.54 (1979), 61–104).