Rank of the Grothendieck group of partial flag varieties

More than a year ago I wrote about dimensions of partial flag varieties, when I was reading Kuznetsov–Polishchuk, Exceptional collections on isotropic Grassmannians. Observe that we always take a maximal parabolic in this setting, as explained in the previous post.

Yesterday I was talking to Maxim Smirnov about these varieties, and we were interested in

• the rank of the Grothendieck group, in other words how long exceptional collections are supposed to be
• the index, i.e. as $\operatorname{Pic}G/P\cong\mathbb{Z}\cdot\mathcal{O}_{G/P}(1)$ the index is defined as the integer $i$ such that $\omega_{G/P}^\vee\cong\mathcal{O}_{G/P}(1)^{\otimes i}$

One can easily compute these numbers from the representation theory associated to these partial flag varieties, but as these numbers are not readily available, let me explain how to compute them, and collect them in a table. One big table is enough per post, so tomorrow you'll get the table for the index.

Rank of the Grothendieck group

This number can be computed as the rank of the homology of $G/P$ using the Bruhat cell decomposition, and is equal to $\#\mathrm{W}_G/\#\mathrm{W}_L$, where $\mathrm{W}_G$ (resp. $\mathrm{W}_L$) denotes the Weyl group of $G$ (resp. $L$), and $L$ is the Levi subgroup inside the maximal parabolic $P$. One can easily compute the (often disconnected) Dynkin diagram associated to $L$ by just removing the chosen vertex for the maximal parabolic. Hence the code is nothing but

def rank(D, k):
  D = DynkinDiagram(D)
  T = [r for r in D.vertices() if r != k]
  WG = D.root_system().root_lattice().weyl_group()
  WL = D.subtype(T).root_system().root_lattice().weyl_group()
  return WG.cardinality() / WL.cardinality()

So if you wanted to find the full exceptional collection on $\mathrm{E}_8/P_4$, good luck with finding almost half a million exceptional objects on this 106-dimensional variety!