# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type F4/P1

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

$\mathrm{F}_{4}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 51 }$

degree
4992
Hilbert series
1, 52, 1053, 12376, 100776, 627912, 3187041, 13748020, 51949755, 175847880, 542393670, 1544927904, 4107092288, 10278624864, 24388573014, 55188666312, 119696471453, 249869263644, 503865726155, 984563860280, ...
Exceptional collections
• Smirnov constructed a full exceptional sequence in 2021, see arXiv:2107.07814.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{F}_{4}/\mathrm{P}_{1})$ are given by:

Homological projective duality