Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow

Using a stability criterion due to Kr"oncke, we show, providing ${n\neq 2k}$,theK\ddot{a}hler--EinsteinmetricontheGrassmannian$Gr_{k}($\mathbb{C}$^{n})$ofcomplex$k$-planesinan$n$-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kr"oncke and Knopf–Sesum on the instability of the Fubini–Study metric on ${\mathbb{C}\mathbb{P}}^n$ for $n>1$. The key to the proof is using the description of Grassmannians as certain coadjoint orbits of $SU(n)$. We are also able to prove that Kr"oncke’s method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.