Hall Polynomials for Nakayama Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k. In this note we first show that the existence of Hall polynomials for \(\mathcal{A}\) equivalent to the existence of the Hall polynomial \(\varphi^{M}_{N L}\) for each \(M, L \in mod\mathcal{A}\) and \(N\in ind\mathcal{A}\). Then we show that for a basic connected Nakayama algebra \(\mathcal{A}\), \(\mathcal{H}(\mathcal{A})=\mathcal{L}(\mathcal{A})\) and Hall polynomials exist for this algebra. We also provide another proof of the existence of Hall polynomials for the representation directed split algebras.