Semi-invariants for tame algebras

Multicoil algebras are characterized by the presence, in their Auslander-Reiten quiver,ofP-families of possible unstable tubes (coils) obtained from stable tubes by a sequence of admissible operations. They are of tame representation type and they exhaust all tame algebras of polynomial growth in case were quire simple connectedness. InthisthesiswestudytheirinvarianttheoryandshowthattheirinniteGITquotientsareproductsofprojectivespaces. Inthesecondinstancewedescribehowthetiltingprocessaffectsthestructureof the rings of semi-invariants.For a hereditary algebra A we identify a subring of the ring of semi-invariants SI(A,d) which can be embedded inside the ring of semi invariants SI(C) of some irreducible component of the variety of e-dimensional modules(forasuitablee)over the tilted algebra End A(T)op. We apply these considerations to the Euclidean case. Finally we analyze the link between some partial orders on the set of orbits of modulesoveranalgebraofniterepresentationtypeandrelationsamongsemiinvariants. We recover the polynomiality of the rings of semi-invariants for tilted algebras of type ⃗ An.