Appears in collection : Symmetry in Geometry and Analysis

Common work with W. Baldoni and M.Walter.

Let $Q = (Q_0, Q_1)$ be a quiver, and $\mathbf n = (n_x)_{x\in Q_0}$ a dimension vector. The group $GL(\mathbf n) = \prod_x GL(n_x)$ acts on the space $H(\mathbf n)$ of representations of $Q$ with dimension vector $\mathbf n$. We give necessary and sufficient inductive conditions of an irreducible representation of $GL(\mathbf n)$ to occur in the space $Sym(H(\mathbf n))$. For a particular quiver, one reobtains the inductive Horn conditions for a representation of $U(\mathbf n)$ to occur in a tensor product of irreducible representations of $U(\mathbf n)$. Paradan results on the decomposition of tensor products of two representations of the holomorphic discrete series of $U(p, q)$ is another consequence.