\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.

Generalised Grassmannian of type E7/P3

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 5 } &= 5 \\ \mathrm{b}_{ 6 } &= 7 \\ \mathrm{b}_{ 7 } &= 10 \\ \mathrm{b}_{ 8 } &= 13 \\ \mathrm{b}_{ 9 } &= 17 \\ \mathrm{b}_{ 10 } &= 22 \\ \mathrm{b}_{ 11 } &= 27 \\ \mathrm{b}_{ 12 } &= 33 \\ \mathrm{b}_{ 13 } &= 39 \\ \mathrm{b}_{ 14 } &= 46 \\ \mathrm{b}_{ 15 } &= 52 \\ \mathrm{b}_{ 16 } &= 60 \\ \mathrm{b}_{ 17 } &= 66 \\ \mathrm{b}_{ 18 } &= 73 \\ \mathrm{b}_{ 19 } &= 78 \\ \mathrm{b}_{ 20 } &= 84 \\ \mathrm{b}_{ 21 } &= 88 \\ \mathrm{b}_{ 22 } &= 92 \\ \mathrm{b}_{ 23 } &= 94 \\ \mathrm{b}_{ 24 } &= 95 \\ \mathrm{b}_{ 25 } &= 95 \\ \mathrm{b}_{ 26 } &= 94 \\ \mathrm{b}_{ 27 } &= 92 \\ \mathrm{b}_{ 28 } &= 88 \\ \mathrm{b}_{ 29 } &= 84 \\ \mathrm{b}_{ 30 } &= 78 \\ \mathrm{b}_{ 31 } &= 73 \\ \mathrm{b}_{ 32 } &= 66 \\ \mathrm{b}_{ 33 } &= 60 \\ \mathrm{b}_{ 34 } &= 52 \\ \mathrm{b}_{ 35 } &= 46 \\ \mathrm{b}_{ 36 } &= 39 \\ \mathrm{b}_{ 37 } &= 33 \\ \mathrm{b}_{ 38 } &= 27 \\ \mathrm{b}_{ 39 } &= 22 \\ \mathrm{b}_{ 40 } &= 17 \\ \mathrm{b}_{ 41 } &= 13 \\ \mathrm{b}_{ 42 } &= 10 \\ \mathrm{b}_{ 43 } &= 7 \\ \mathrm{b}_{ 44 } &= 5 \\ \mathrm{b}_{ 45 } &= 3 \\ \mathrm{b}_{ 46 } &= 2 \\ \mathrm{b}_{ 47 } &= 1 \\ \mathrm{b}_{ 48 } &= 1 \end{align*}
Basic information
dimension
47
index
11
Euler characteristic
2016
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 5$, $\mathrm{b}_{ 6 } = 7$, $\mathrm{b}_{ 7 } = 10$, $\mathrm{b}_{ 8 } = 13$, $\mathrm{b}_{ 9 } = 17$, $\mathrm{b}_{ 10 } = 22$, $\mathrm{b}_{ 11 } = 27$, $\mathrm{b}_{ 12 } = 33$, $\mathrm{b}_{ 13 } = 39$, $\mathrm{b}_{ 14 } = 46$, $\mathrm{b}_{ 15 } = 52$, $\mathrm{b}_{ 16 } = 60$, $\mathrm{b}_{ 17 } = 66$, $\mathrm{b}_{ 18 } = 73$, $\mathrm{b}_{ 19 } = 78$, $\mathrm{b}_{ 20 } = 84$, $\mathrm{b}_{ 21 } = 88$, $\mathrm{b}_{ 22 } = 92$, $\mathrm{b}_{ 23 } = 94$, $\mathrm{b}_{ 24 } = 95$, $\mathrm{b}_{ 25 } = 95$, $\mathrm{b}_{ 26 } = 94$, $\mathrm{b}_{ 27 } = 92$, $\mathrm{b}_{ 28 } = 88$, $\mathrm{b}_{ 29 } = 84$, $\mathrm{b}_{ 30 } = 78$, $\mathrm{b}_{ 31 } = 73$, $\mathrm{b}_{ 32 } = 66$, $\mathrm{b}_{ 33 } = 60$, $\mathrm{b}_{ 34 } = 52$, $\mathrm{b}_{ 35 } = 46$, $\mathrm{b}_{ 36 } = 39$, $\mathrm{b}_{ 37 } = 33$, $\mathrm{b}_{ 38 } = 27$, $\mathrm{b}_{ 39 } = 22$, $\mathrm{b}_{ 40 } = 17$, $\mathrm{b}_{ 41 } = 13$, $\mathrm{b}_{ 42 } = 10$, $\mathrm{b}_{ 43 } = 7$, $\mathrm{b}_{ 44 } = 5$, $\mathrm{b}_{ 45 } = 3$, $\mathrm{b}_{ 46 } = 2$, $\mathrm{b}_{ 47 } = 1$, $\mathrm{b}_{ 48 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{3})$
adjoint group of type $\mathrm{E}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{3})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{3})$
133
Projective geometry
minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{3}\hookrightarrow\mathbb{P}^{ 8644 }$

degree
135848348778713395019120640
Hilbert series
1, 8645, 13728792, 7323895800, 1778618202669, 236623544427270, 19519433726417625, 1090168539699417750, 43993460888906665500, 1348095176669475954720, 32611758857289095218176, 642443206178278977561600, 10569578762996426893513248, 148276471308422875531385580, 1804732963658859170305997394, 19339069897951186534147208905, 184736647641066570617069001500, 1590064141215173345940892818375, 12446412671545323701870264625000, 89319908088757664551347148125000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{7}/\mathrm{P}_{3})$. Will you be the first to construct one? Let us know if you do!