\begin{equation} \DeclareMathOperator\Gr{Gr} \DeclareMathOperator\LGr{LGr} \DeclareMathOperator\OGr{OGr} \DeclareMathOperator\SGr{SGr} \DeclareMathOperator\Kzero{K_0} \DeclareMathOperator\index{i} \DeclareMathOperator\rk{rk} \end{equation}

Grassmannian.info

A periodic table of (generalised) Grassmannians.

Orthogonal Grassmannian $\OGr(2,8)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 3 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 5 } &= 4 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 7 } &= 3 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 9 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \end{align*}
Basic information
dimension
9
index
5
Euler characteristic
24
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 3$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 4$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 3$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 9 } = 1$, $\mathrm{b}_{ 10 } = 1$
$\mathrm{Aut}^0(\OGr(2,8))$
$\mathrm{PSO}_{ 8 }$
$\pi_0\mathrm{Aut}(\OGr(2,8))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\OGr(2,8))$
28
Projective geometry
minimal embedding

$\OGr(2,8)\hookrightarrow\mathbb{P}^{ 27 }$

degree
168
Hilbert series
1, 28, 300, 1925, 8918, 32928, 102816, 282150, 698775, 1591876, 3383380, 6782139, 12931100, 23609600, 41505024, 70570332, 116486397, 187250700, 293916700, 451511137, ...
Exceptional collections
  • Kuznetsov–Smirnov constructed a full exceptional sequence in 2020, see arXiv:2001.04148.