# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Orthogonal Grassmannian $\OGr(2,11)$

Basic information
dimension
15
index
8
Euler characteristic
40
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 18 } = 4$, $\mathrm{b}_{ 20 } = 3$, $\mathrm{b}_{ 22 } = 3$, $\mathrm{b}_{ 24 } = 2$, $\mathrm{b}_{ 26 } = 2$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$
$\mathrm{Aut}^0(\OGr(2,11))$
$\mathrm{SO}_{ 11 }$
$\pi_0\mathrm{Aut}(\OGr(2,11))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,11))$
55
Projective geometry
minimal embedding

$\OGr(2,11)\hookrightarrow\mathbb{P}^{ 54 }$

degree
5720
Hilbert series
1, 55, 1144, 13650, 112200, 703494, 3586440, 15520791, 58790875, 199377750, 615879264, 1756322360, 4673587776, 11705713800, 27793245600, 62928540246, 136548771645, 285166559909, 575248883000, 1124398096250, ...
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 $\mathrm{c}_1(\OGr(2,11))$ are given by:

Homological projective duality