On some Ringel–Hall products in tame cases

Let k be a finite field and consider the finite dimensional path algebra kQ where Q is a quiver of tame type, i.e. of type (non-cyclic), . On one hand we describe the Ringel–Hall product [I][P] where I is an indecomposable preinjective of defect 1 and P an indecomposable preprojective of defect −1. We generalize in this way a formula by Hubery in [7]. On the other hand we determine the products [R][P] (and dually [I][R]) for P an indecomposable preprojective of defect −1 and R a regular module from a homogeneous tube or a regular quasi-semisimple module from a non-homogeneous tube. Some applications are also shown.