Hall polynomials and composition algebra of representation finite algebras

Abstract

Let A be a representation finite algebra over finite field k such that the indecomposable A-modules are determined by their dimension vectors and for each M, L ind(A) and N mod(A), either FMN L=0 or F ML N=0. We show that A has Hall polynomials and the rational extension of its Ringel-Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if A is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in mod(A), or representation finite cluster tilted algebra, then A has Hall polynomials and H (A) ℤ Q=C (A)ℤQ. © 2013 Springer Science+Business Media Dordrecht.