  In this paper, we show that the deformed Hermitian Yang-Mills (dHYM) equation
on a rational homogeneous variety, equipped with any invariant K\"{a}hler
metric, always admits a solution. In particular, we describe the Lagrangian
phase, with respect to any invariant K\"{a}hler metric, of every closed
invariant $(1,1)$-form in terms of Lie theory. Building on this, we
characterize all supercritical and hypercritical homogeneous solutions of the
dHYM equation using the Cartan matrix associated with the underlying complex
simple Lie algebra. Further, we provide an explicit formula, in terms of Lie
theory, for the slope of holomorphic vector bundles over rational homogeneous
varieties. Using this formula, we derive a new criterion for slope
semistability through restrictions of holomorphic vector bundles to the
generators of the associated cone of curves. Moreover, we provide a new
characterization, in terms of central charges defined by rational curves, for
slope (semi)stability of holomorphic vector bundles over rational homogeneous
varieties. As an application of our main results, we describe all supercritical
and hypercritical homogeneous solutions of the dHYM equation on the Fano
threefold defined by the Wallach flag manifold
${\mathbb{P}}(T_{{\mathbb{P}^{2}}})$. In addition, we introduce a constructive
method to obtain non-trivial examples of Hermitian-Einstein metrics on certain
holomorphic vector bundles over ${\mathbb{P}}(T_{{\mathbb{P}^{2}}})$ from
solutions of linear diophantine equations. In this last case, we describe
explicitly all associated Hermitian Yang-Mills instantons. We also present some
new insights that explore the interplay between intersection theory and number
theory.
