Einstein metrics on the full flag $F(n)$

Let $M=G\mathrm{/}K$ be a full flag manifold. In this work, we investigate the $G$-stability of Einstein metrics on $M$ and analyze their stability types, including coindices, for several cases. We specifically focus on $F(n)=\mathrm{S}\mathrm{U}(n)\mathrm{/}T$, emphasizing $n=5$, where we identify four new Einstein metrics in addition to known ones. Stability data, including coindex and Hessian spectrum, confirms that these metrics on $F(5)$ are pairwise non-homothetic, providing new insights into the finiteness conjecture.