\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 $\OGr(2,15)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 3 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 7 } &= 4 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 5 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 11 } &= 6 \\ \mathrm{b}_{ 12 } &= 6 \\ \mathrm{b}_{ 13 } &= 6 \\ \mathrm{b}_{ 14 } &= 6 \\ \mathrm{b}_{ 15 } &= 5 \\ \mathrm{b}_{ 16 } &= 5 \\ \mathrm{b}_{ 17 } &= 4 \\ \mathrm{b}_{ 18 } &= 4 \\ \mathrm{b}_{ 19 } &= 3 \\ \mathrm{b}_{ 20 } &= 3 \\ \mathrm{b}_{ 21 } &= 2 \\ \mathrm{b}_{ 22 } &= 2 \\ \mathrm{b}_{ 23 } &= 1 \\ \mathrm{b}_{ 24 } &= 1 \end{align*}
Basic information
dimension
23
index
12
Euler characteristic
84
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 5$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 11 } = 6$, $\mathrm{b}_{ 12 } = 6$, $\mathrm{b}_{ 13 } = 6$, $\mathrm{b}_{ 14 } = 6$, $\mathrm{b}_{ 15 } = 5$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 17 } = 4$, $\mathrm{b}_{ 18 } = 4$, $\mathrm{b}_{ 19 } = 3$, $\mathrm{b}_{ 20 } = 3$, $\mathrm{b}_{ 21 } = 2$, $\mathrm{b}_{ 22 } = 2$, $\mathrm{b}_{ 23 } = 1$, $\mathrm{b}_{ 24 } = 1$
$\mathrm{Aut}^0(\OGr(2,15))$
$\mathrm{SO}_{ 15 }$
$\pi_0\mathrm{Aut}(\OGr(2,15))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,15))$
105
Projective geometry
minimal embedding

$\OGr(2,15)\hookrightarrow\mathbb{P}^{ 104 }$

degree
832048
Hilbert series
1, 105, 4080, 88179, 1270815, 13537524, 113859200, 791224200, 4695002676, 24385860460, 113015849856, 474507071220, 1827174287820, 6518164602264, 21722800137600, 68110029598100, 202120945079625, 570619024282125, 1539370437368400, 3983638152414375, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.