Hall Polynomials For Affine Hereditary Algebras Over Finite Fields

Let(Q,?)be a finite quiver Q with an automorphism ?.It is known that the pair(Q,?)defines a family of algebras(?)(Q,?;q)over finite fields Fq of q elements.In this dissertation we study Hall polynomials for the algebras(?)(Q,?;q)under the assumption that Q is an affine quiver.We first define the notion of decomposition sequences for(Q,?)which parameterize isoclasses(=isomorphism classes)of finite dimensional modules over(?)(Q,?;q).Our main purpose is to prove that Hall polynomial ??,??(t)exists for each triple of decomposition sequences ?,?,? of same type for(Q,?).If ? is an identity automorphism,we recover the main results in both Hubery[1]and Deng-Ruan[2].Moreover,in comparison with Deng-Ruan's method,our proof is much simpler without passing over the category of coherent sheaves on the corresponding weighted projective line and without use of the derived equivalence.The proof of our main theorem involves combinatorial and homological methods,based on the structure and homological properties of the module categories over affine hereditary algebras.More precisely,we proceed induction on dimension vectors,as well as on defects of indecomposable postprojective and preinjective modules.We first prove that several sums of Hall numbers and their products are given by polynomials under some induction hypothesis,and then reduce the general case to some particular triples of decomposition sequences by making use of the associativity of Hall algebras and Green's formula.We finally complete the proof of the main theorem by induction on the defect.As an application,we prove that Hall polynomials exist for coherent sheaves on a domestic weighted projective line,as well as in the case of those with weight permutation.