Grassmannian.info

A periodic table of (generalised) Grassmannians.

Freudenthal variety

Basic information
dimension
27
index
18
Euler characteristic
56
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 3$, $\mathrm{b}_{ 20 } = 3$, $\mathrm{b}_{ 22 } = 3$, $\mathrm{b}_{ 24 } = 3$, $\mathrm{b}_{ 26 } = 3$, $\mathrm{b}_{ 28 } = 3$, $\mathrm{b}_{ 30 } = 3$, $\mathrm{b}_{ 32 } = 3$, $\mathrm{b}_{ 34 } = 3$, $\mathrm{b}_{ 36 } = 3$, $\mathrm{b}_{ 38 } = 2$, $\mathrm{b}_{ 40 } = 2$, $\mathrm{b}_{ 42 } = 2$, $\mathrm{b}_{ 44 } = 2$, $\mathrm{b}_{ 46 } = 1$, $\mathrm{b}_{ 48 } = 1$, $\mathrm{b}_{ 50 } = 1$, $\mathrm{b}_{ 52 } = 1$, $\mathrm{b}_{ 54 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{7})$
adjoint group of type $\mathrm{E}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{7})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{7})$
133
Projective geometry
minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{7}\hookrightarrow\mathbb{P}^{ 55 }$

degree
13110
Hilbert series
1, 56, 1463, 24320, 293930, 2785552, 21737254, 144538624, 839848450, 4347450800, 20355385710, 87265194240, 345992859975, 1279301331000, 4442249264625, 14573017267200, 45398364338250, 134897996890800, 383822534859750, 1049290591104000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{7}/\mathrm{P}_{7})$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{7}/\mathrm{P}_{7})$ are given by:

Homological projective duality