Residual categories for (co)adjoint Grassmannians in classical types

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types An and Dn, that is, flag varieties Fl(1, n; n + 1) and isotropic orthogonal Grassmannians OG(2, 2n); in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For OG(2, 2n) this is the first exceptional collection proved to be full.

Automated summary

In this study, we extend and provide support for a conjecture regarding the relationship between the small quantum cohomology ring and derived category of coherent sheaves of a smooth Fano variety. By examining specific examples, we demonstrate the validity of this conjecture in the context of (co)adjoint homogeneous varieties of simple algebraic groups.