Quivers, long exact sequences and Horn type inequalities

We give necessary and sufficient inequalities for the existence of long exact sequences of m finite abelian p-groups with fixed isomorphism types. This problem is related to some generalized Littlewood–Richardson coefficients that we define in this paper. We also show how this problem is related to eigenvalues of Hermitian matrices satisfying certain (in)equalities. When m=3, we recover the Horn type inequalities that solve the saturation conjecture for Littlewood–Richardson coefficients and Horn's conjecture.