# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Projective space $\mathbb{P}^{11}$

There exist other realisations of this Grassmannian:
Basic information
dimension
11
index
12
Euler characteristic
12
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 22 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{11})$
$\mathrm{PGL}_{ 12 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{11})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{11})$
143
Projective geometry
minimal embedding

$\mathbb{P}^{11}\hookrightarrow\mathbb{P}^{ 11 }$

More appropriately in this particular presentation as a quotient of the symplectic group is to consider the Grassmannian as the adjoint variety of type $\mathrm{C}_{ 6 }$, where the embedding is the second Veronese embedding into $\mathbb{P}(\mathrm{V}_{2\omega_1})=\mathbb{P}^{ 77 }$.

degree
1
Hilbert series
1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, ...
Exceptional collections
• Beilinson constructed a full exceptional sequence in 1978, see MR0509388.
• Kapranov constructed a full exceptional sequence in 1988, see MR0939472.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathbb{P}^{11})$ are given by:

Homological projective duality