# An update for Grassmannian.info: exceptional collections

Lockdown is good for developing mathematical websites, I guess. I've pushed another update to **grassmannian.info**: it now shows some basic information about the derived categories of generalised Grassmannians, namely when it is known that a full exceptional collection exists. An important folklore conjecture states that this is the case, but as you can see when you enable the coloring in the table, we are still a long way off of settling it.

What I never realised is that, out of all known cases, **only one** is not (co)minuscule or (co)adjoint. These are somehow the easiest varieties, and for the majority of them (see below) a full exceptional collection is known. The unique example outside this class is constructed by Guseva in On the derived category of $\mathrm{IGr}(3,8)$ (denoted $\mathrm{SGr}(3,8)$ on grassmannian.info.

Out of the (co)minuscule or (co)adjoint varieties, the ones for which an exceptional collection is not yet constructed are:

- the (co)adjoint in type E
_{6}: E_{6}/P_{2} - the (co)minuscule in type E
_{7}: the Freudenthal variety, or E_{7}/P_{7} - the (co)adjoint in type E
_{7}: E_{7}/P_{1} - the (co)adjoint in type E
_{8}: E_{8}/P_{8} - the adjoint in type F
_{4}: F_{4}/P_{1}

If you manage to construct an exceptional collection in one of these cases, I will give you a large bar of your preferred Belgian chocolate, if you need some further incentive to think about it.

I've also completed copying the information from Bourbaki's appendix.