Derived Equivalences of Generalized Grassmannian Flops: $D_4$ and $G_2^{†}$ Cases

We prove that the generalized Grassmannian flops of both $D_4$ and $G_2^{†}$ type induce derived equivalences, which provide new evidence for the DK conjecture by Bondal-Orlov and Kawamta. The proof is based on Kuznetsov’s mutation technique, which takes a sequence of mutations of exceptional objects.