# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Grassmannian $\Gr(5,8)$

There exist other realisations of this Grassmannian:
Basic information
dimension
15
index
8
Euler characteristic
56
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 6$, $\mathrm{b}_{ 14 } = 6$, $\mathrm{b}_{ 16 } = 6$, $\mathrm{b}_{ 18 } = 6$, $\mathrm{b}_{ 20 } = 5$, $\mathrm{b}_{ 22 } = 4$, $\mathrm{b}_{ 24 } = 3$, $\mathrm{b}_{ 26 } = 2$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$
$\mathrm{Aut}^0(\Gr(5,8))$
$\mathrm{PGL}_{ 8 }$
$\pi_0\mathrm{Aut}(\Gr(5,8))$
$1$
$\dim\mathrm{Aut}^0(\Gr(5,8))$
63
Projective geometry
minimal embedding

$\Gr(5,8)\hookrightarrow\mathbb{P}^{ 55 }$

degree
6006
Hilbert series
1, 56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480, 1178943365600, ...
Exceptional collections
• 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 $\mathrm{c}_1(\Gr(5,8))$ are given by:

Homological projective duality