The b-functions of quiver semi-invariants
We introduce two techniques that will allow us to study recursively the $b$-functions of various semi-invariants. The main application will be for semi-invariants of quivers, with emphasis on Dynkin quivers. The first method will be based on a slice technique, while the second on a relation of $b$-functions under castling transforms. The latter will enable us to compute all the $b$-functions of Dynkin quivers. These techniques are also useful for other reasons: they provide natural isomorphisms between rings of semi-invariants, they can describe the locally semi-simple representation corresponding to a semi-invariant and give the necessary inequalities between the values of the dimension vector for the semi-invariant to be non-zero. Additionally, for quivers of type $D_n$ we give a uniform rule for the explicit description of the generic element of a representation space corresponding to a dimension vector. We provide several examples to show how all computations can be carried out by hand.