# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Projective space $\mathbb{P}^{13}$

There exist other realisations of this Grassmannian:
Basic information
dimension
13
index
14
Euler characteristic
14
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 22 } = 1$, $\mathrm{b}_{ 24 } = 1$, $\mathrm{b}_{ 26 } = 1$
$\mathrm{Aut}^0(\mathbb{P}^{13})$
$\mathrm{PGL}_{ 14 }$
$\pi_0\mathrm{Aut}(\mathbb{P}^{13})$
$1$
$\dim\mathrm{Aut}^0(\mathbb{P}^{13})$
195
Projective geometry
minimal embedding

$\mathbb{P}^{13}\hookrightarrow\mathbb{P}^{ 13 }$

More appropriately in this particular presentation as a quotient of the symplectic group is to consider the Grassmannian as the adjoint variety of type $\mathrm{C}_{ 7 }$, where the embedding is the second Veronese embedding into $\mathbb{P}(\mathrm{V}_{2\omega_1})=\mathbb{P}^{ 104 }$.

degree
1
Hilbert series
1, 14, 105, 560, 2380, 8568, 27132, 77520, 203490, 497420, 1144066, 2496144, 5200300, 10400600, 20058300, 37442160, 67863915, 119759850, 206253075, 347373600, ...
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 $\mathrm{c}_1(\mathbb{P}^{13})$ are given by:

Homological projective duality