\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}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 5 \\ \mathrm{b}_{ 10 } &= 7 \\ \mathrm{b}_{ 12 } &= 10 \\ \mathrm{b}_{ 14 } &= 13 \\ \mathrm{b}_{ 16 } &= 17 \\ \mathrm{b}_{ 18 } &= 22 \\ \mathrm{b}_{ 20 } &= 27 \\ \mathrm{b}_{ 22 } &= 33 \\ \mathrm{b}_{ 24 } &= 39 \\ \mathrm{b}_{ 26 } &= 46 \\ \mathrm{b}_{ 28 } &= 52 \\ \mathrm{b}_{ 30 } &= 60 \\ \mathrm{b}_{ 32 } &= 66 \\ \mathrm{b}_{ 34 } &= 73 \\ \mathrm{b}_{ 36 } &= 78 \\ \mathrm{b}_{ 38 } &= 84 \\ \mathrm{b}_{ 40 } &= 88 \\ \mathrm{b}_{ 42 } &= 92 \\ \mathrm{b}_{ 44 } &= 94 \\ \mathrm{b}_{ 46 } &= 95 \\ \mathrm{b}_{ 48 } &= 95 \\ \mathrm{b}_{ 50 } &= 94 \\ \mathrm{b}_{ 52 } &= 92 \\ \mathrm{b}_{ 54 } &= 88 \\ \mathrm{b}_{ 56 } &= 84 \\ \mathrm{b}_{ 58 } &= 78 \\ \mathrm{b}_{ 60 } &= 73 \\ \mathrm{b}_{ 62 } &= 66 \\ \mathrm{b}_{ 64 } &= 60 \\ \mathrm{b}_{ 66 } &= 52 \\ \mathrm{b}_{ 68 } &= 46 \\ \mathrm{b}_{ 70 } &= 39 \\ \mathrm{b}_{ 72 } &= 33 \\ \mathrm{b}_{ 74 } &= 27 \\ \mathrm{b}_{ 76 } &= 22 \\ \mathrm{b}_{ 78 } &= 17 \\ \mathrm{b}_{ 80 } &= 13 \\ \mathrm{b}_{ 82 } &= 10 \\ \mathrm{b}_{ 84 } &= 7 \\ \mathrm{b}_{ 86 } &= 5 \\ \mathrm{b}_{ 88 } &= 3 \\ \mathrm{b}_{ 90 } &= 2 \\ \mathrm{b}_{ 92 } &= 1 \\ \mathrm{b}_{ 94 } &= 1 \end{align*}
Basic information
dimension
47
index
11
Euler characteristic
2016
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 7$, $\mathrm{b}_{ 12 } = 10$, $\mathrm{b}_{ 14 } = 13$, $\mathrm{b}_{ 16 } = 17$, $\mathrm{b}_{ 18 } = 22$, $\mathrm{b}_{ 20 } = 27$, $\mathrm{b}_{ 22 } = 33$, $\mathrm{b}_{ 24 } = 39$, $\mathrm{b}_{ 26 } = 46$, $\mathrm{b}_{ 28 } = 52$, $\mathrm{b}_{ 30 } = 60$, $\mathrm{b}_{ 32 } = 66$, $\mathrm{b}_{ 34 } = 73$, $\mathrm{b}_{ 36 } = 78$, $\mathrm{b}_{ 38 } = 84$, $\mathrm{b}_{ 40 } = 88$, $\mathrm{b}_{ 42 } = 92$, $\mathrm{b}_{ 44 } = 94$, $\mathrm{b}_{ 46 } = 95$, $\mathrm{b}_{ 48 } = 95$, $\mathrm{b}_{ 50 } = 94$, $\mathrm{b}_{ 52 } = 92$, $\mathrm{b}_{ 54 } = 88$, $\mathrm{b}_{ 56 } = 84$, $\mathrm{b}_{ 58 } = 78$, $\mathrm{b}_{ 60 } = 73$, $\mathrm{b}_{ 62 } = 66$, $\mathrm{b}_{ 64 } = 60$, $\mathrm{b}_{ 66 } = 52$, $\mathrm{b}_{ 68 } = 46$, $\mathrm{b}_{ 70 } = 39$, $\mathrm{b}_{ 72 } = 33$, $\mathrm{b}_{ 74 } = 27$, $\mathrm{b}_{ 76 } = 22$, $\mathrm{b}_{ 78 } = 17$, $\mathrm{b}_{ 80 } = 13$, $\mathrm{b}_{ 82 } = 10$, $\mathrm{b}_{ 84 } = 7$, $\mathrm{b}_{ 86 } = 5$, $\mathrm{b}_{ 88 } = 3$, $\mathrm{b}_{ 90 } = 2$, $\mathrm{b}_{ 92 } = 1$, $\mathrm{b}_{ 94 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{3})$
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!

Quantum cohomology

The small quantum cohomology is not generically semisimple.

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

Homological projective duality