This blogpost does not serve as an introduction to Dubrovin's conjecture, let me just say that it states the equivalence between

• the existence of a full exceptional collection in $\mathbf{D}^{\mathrm{b}}(G/P)$
• the generic semisimplicity of (big) quantum cohomology $\mathrm{BQH}(G/P)$

The first is a statement about the algebraic geometry of $G/P$, whilst the second concerns the symplectic geometry of $G/P$, and their equivalence is an amazing prediction motivated by mirror symmetry.

One thing we can do is tabulate for which $G/P$ either is known to hold, and Grassmannian.info now does that. Remark that, whilst the conjecture concerns the generic semisimplicity of big quantum cohomology, there is an easier version called small quantum cohomology, where computations are more tractable. And semisimplicity of the small version implies it for the big version, but it is not equivalent to it as there can be cases where the small quantum cohomology is not semisimple.

Let me just end by saying that the data on the website is based on the table on page 33 of Chaput–Perrin: On the quantum cohomology of adjoint varieties and for the generic semisimplicity of the coadjoint Grassmannians on unpublished work-in-progress of Chaput–Smirnov. I want to thank Maxim Smirnov for interesting discussions on the subject.

The conjecture is phrased for all smooth projective varieties, but for the purposes of this blogpost we only care about $G/P$.