Balanced metrics on homogeneous vector bundles

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M,\omega )$ and let $E=E_1\oplus \mathrm{.}\mathrm{.}\mathrm{.}\oplus E_m\to M$ be its decomposition into irreducible factors. Suppose that each $E_j$ admits a $\omega$-balanced metric in Donaldson-Wang terminology. In this paper we prove that $E$ admits a unique $\omega$-balanced metric if and only if $\frac{r_j}{N_j}=\frac{r_k}{N_k}$ for all $j,k=1,\mathrm{.}\mathrm{.}\mathrm{.},m$, where $r_j$ denotes the rank of $E_j$ and $N_j=\dim H^0(M,E_j)$. We apply our result to the case of homogeneous vector bundles over a rational homogeneous variety $(M,\omega )$ and we show the existence and rigidity of balanced Kaehler embedding from $(M, \omega)$ into Grassmannians.