# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Orthogonal Grassmannian $\OGr(2,13)$

Basic information
dimension
19
index
10
Euler characteristic
60
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 5$, $\mathrm{b}_{ 22 } = 5$, $\mathrm{b}_{ 24 } = 4$, $\mathrm{b}_{ 26 } = 4$, $\mathrm{b}_{ 28 } = 3$, $\mathrm{b}_{ 30 } = 3$, $\mathrm{b}_{ 32 } = 2$, $\mathrm{b}_{ 34 } = 2$, $\mathrm{b}_{ 36 } = 1$, $\mathrm{b}_{ 38 } = 1$
$\mathrm{Aut}^0(\OGr(2,13))$
$\mathrm{SO}_{ 13 }$
$\pi_0\mathrm{Aut}(\OGr(2,13))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,13))$
78
Projective geometry
minimal embedding

$\OGr(2,13)\hookrightarrow\mathbb{P}^{ 77 }$

degree
67184
Hilbert series
1, 78, 2275, 37400, 415701, 3461458, 23046023, 128271000, 616258500, 2617486872, 10011037452, 34981096384, 112977874100, 340467258600, 964904283772, 2588633683064, 6610679343345, 16146190238750, 37871772241875, 85609271147400, ...
Exceptional collections
• Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(2,13))$ are given by:

Homological projective duality