THE HALL POLYNOMIALS OF A CYCLIC SERIAL ALGEBRA

Hall algebra is constructed from the finite modules of a finitary algebra and is strongly related to the the studies of quantum groups. C.M.Ringel introduces the Hall algebra of a finitary algebra and studies the structures of Hall algebras and their relationship with the quantized enveloping algebra of Lie algebras and Lusztig uses the methods of Hall algebras to construct the canonical basis of a quantized enveloping algebra of some Lie algebras([2],[4]). Let R be a finitary ring and F-M1,...,Ml be the number of filtrations L=L?0 superset of L?1...superset of L?l=0 such that L?(t-1(L)/L(t) congruent to M(t) for t = 1,...,l, then F-M,N(L) is the number of submodules L' of a finite module L of R with the property that L' congruent to N and L/L' congruent to M. The Hall algebra H(R) is a free abelian group with a basis {u([M])}([M]) indexed by the isomorphism classes of finite R modules with the multiplication defined by [GRAPHICS] It is proved in [2] that the structural constants F-M,N(L) for a representation directed algebra R over a finite field k can be get from integral polynomials, called Hall polynomials, by taking value at the cardinal of k. In [4], he proves a similar results for F-S1,...,St(M) for a tube over a finite field. He also calculates these polynomials for the representation finite hereditary algebras in [3]. It is not easy to calculate the Hall polynomials in general. In this paper, we find a method to calculate F-M,N(L) for R to be a cyclic serial algebra, we give a formula for the case when N is semisimple, and using an order defined in [1], this leads to a inductive process to calculating F-M,N(L). The results are also used to prove the existence of Hall polynomials for such algebras. Furthermore, our method of calculating the Hall polynomial also valid for the case of a tube.