Grassmannian.info

A periodic table of (generalised) Grassmannians.

Generalised Grassmannian of type F4/P4

Basic information
dimension
15
index
11
Euler characteristic
24
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 2$, $\mathrm{b}_{ 22 } = 2$, $\mathrm{b}_{ 24 } = 1$, $\mathrm{b}_{ 26 } = 1$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$
$\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{4})$
$\mathrm{F}_4$
$\pi_0\mathrm{Aut}(\mathrm{F}_{4}/\mathrm{P}_{4})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{4})$
52
Projective geometry
minimal embedding

$\mathrm{F}_{4}/\mathrm{P}_{4}\hookrightarrow\mathbb{P}^{ 25 }$

degree
78
Hilbert series
1, 26, 324, 2652, 16302, 81081, 342056, 1264120, 4188834, 12664184, 35405968, 92512368, 227854536, 532703874, 1189056024, 2546364040, 5253305915, 10477865970, 20265831300, 38111646300, ...
Exceptional collections
• Belmans–Kuznetsov–Smirnov constructed a full exceptional sequence in 2020, see arXiv:2005.01989.
Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{F}_{4}/\mathrm{P}_{4})$ are given by:

Homological projective duality