# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Orthogonal Grassmannian $\OGr(4,12)$

Basic information
dimension
22
index
7
Euler characteristic
240
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 8 } = 7$, $\mathrm{b}_{ 10 } = 9$, $\mathrm{b}_{ 12 } = 12$, $\mathrm{b}_{ 14 } = 15$, $\mathrm{b}_{ 16 } = 17$, $\mathrm{b}_{ 18 } = 20$, $\mathrm{b}_{ 20 } = 20$, $\mathrm{b}_{ 22 } = 22$, $\mathrm{b}_{ 24 } = 20$, $\mathrm{b}_{ 26 } = 20$, $\mathrm{b}_{ 28 } = 17$, $\mathrm{b}_{ 30 } = 15$, $\mathrm{b}_{ 32 } = 12$, $\mathrm{b}_{ 34 } = 9$, $\mathrm{b}_{ 36 } = 7$, $\mathrm{b}_{ 38 } = 4$, $\mathrm{b}_{ 40 } = 3$, $\mathrm{b}_{ 42 } = 1$, $\mathrm{b}_{ 44 } = 1$
$\mathrm{Aut}^0(\OGr(4,12))$
$\mathrm{PSO}_{ 12 }$
$\pi_0\mathrm{Aut}(\OGr(4,12))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\OGr(4,12))$
66
Projective geometry
minimal embedding

$\OGr(4,12)\hookrightarrow\mathbb{P}^{ 494 }$

degree
5462865408
Hilbert series
1, 495, 55055, 2550548, 65493792, 1095915744, 13232722140, 123515648685, 935877829315, 5967356119725, 32906870606610, 160314212254560, 701733465072640, 2797750569360384, 10273887744211872, 35073296276201118, 112179553015334805, 338384405311947995, 968089306006700801, 2639608173710364780, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\OGr(4,12))$. Will you be the first to construct one? Let us know if you do!

Kuznetsov–Polishchuk have constructed an exceptional collection of maximal length in MR3463417. Can you prove it's full?

Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is not known yet to be generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(4,12))$ are given by:

Homological projective duality