\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/P4

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 3 \\ \mathrm{b}_{ 4 } &= 5 \\ \mathrm{b}_{ 5 } &= 9 \\ \mathrm{b}_{ 6 } &= 13 \\ \mathrm{b}_{ 7 } &= 21 \\ \mathrm{b}_{ 8 } &= 29 \\ \mathrm{b}_{ 9 } &= 41 \\ \mathrm{b}_{ 10 } &= 55 \\ \mathrm{b}_{ 11 } &= 72 \\ \mathrm{b}_{ 12 } &= 92 \\ \mathrm{b}_{ 13 } &= 115 \\ \mathrm{b}_{ 14 } &= 141 \\ \mathrm{b}_{ 15 } &= 168 \\ \mathrm{b}_{ 16 } &= 200 \\ \mathrm{b}_{ 17 } &= 230 \\ \mathrm{b}_{ 18 } &= 264 \\ \mathrm{b}_{ 19 } &= 295 \\ \mathrm{b}_{ 20 } &= 329 \\ \mathrm{b}_{ 21 } &= 357 \\ \mathrm{b}_{ 22 } &= 387 \\ \mathrm{b}_{ 23 } &= 410 \\ \mathrm{b}_{ 24 } &= 432 \\ \mathrm{b}_{ 25 } &= 447 \\ \mathrm{b}_{ 26 } &= 459 \\ \mathrm{b}_{ 27 } &= 464 \\ \mathrm{b}_{ 28 } &= 464 \\ \mathrm{b}_{ 29 } &= 459 \\ \mathrm{b}_{ 30 } &= 447 \\ \mathrm{b}_{ 31 } &= 432 \\ \mathrm{b}_{ 32 } &= 410 \\ \mathrm{b}_{ 33 } &= 387 \\ \mathrm{b}_{ 34 } &= 357 \\ \mathrm{b}_{ 35 } &= 329 \\ \mathrm{b}_{ 36 } &= 295 \\ \mathrm{b}_{ 37 } &= 264 \\ \mathrm{b}_{ 38 } &= 230 \\ \mathrm{b}_{ 39 } &= 200 \\ \mathrm{b}_{ 40 } &= 168 \\ \mathrm{b}_{ 41 } &= 141 \\ \mathrm{b}_{ 42 } &= 115 \\ \mathrm{b}_{ 43 } &= 92 \\ \mathrm{b}_{ 44 } &= 72 \\ \mathrm{b}_{ 45 } &= 55 \\ \mathrm{b}_{ 46 } &= 41 \\ \mathrm{b}_{ 47 } &= 29 \\ \mathrm{b}_{ 48 } &= 21 \\ \mathrm{b}_{ 49 } &= 13 \\ \mathrm{b}_{ 50 } &= 9 \\ \mathrm{b}_{ 51 } &= 5 \\ \mathrm{b}_{ 52 } &= 3 \\ \mathrm{b}_{ 53 } &= 1 \\ \mathrm{b}_{ 54 } &= 1 \end{align*}
Basic information
dimension
53
index
8
Euler characteristic
10080
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 3$, $\mathrm{b}_{ 4 } = 5$, $\mathrm{b}_{ 5 } = 9$, $\mathrm{b}_{ 6 } = 13$, $\mathrm{b}_{ 7 } = 21$, $\mathrm{b}_{ 8 } = 29$, $\mathrm{b}_{ 9 } = 41$, $\mathrm{b}_{ 10 } = 55$, $\mathrm{b}_{ 11 } = 72$, $\mathrm{b}_{ 12 } = 92$, $\mathrm{b}_{ 13 } = 115$, $\mathrm{b}_{ 14 } = 141$, $\mathrm{b}_{ 15 } = 168$, $\mathrm{b}_{ 16 } = 200$, $\mathrm{b}_{ 17 } = 230$, $\mathrm{b}_{ 18 } = 264$, $\mathrm{b}_{ 19 } = 295$, $\mathrm{b}_{ 20 } = 329$, $\mathrm{b}_{ 21 } = 357$, $\mathrm{b}_{ 22 } = 387$, $\mathrm{b}_{ 23 } = 410$, $\mathrm{b}_{ 24 } = 432$, $\mathrm{b}_{ 25 } = 447$, $\mathrm{b}_{ 26 } = 459$, $\mathrm{b}_{ 27 } = 464$, $\mathrm{b}_{ 28 } = 464$, $\mathrm{b}_{ 29 } = 459$, $\mathrm{b}_{ 30 } = 447$, $\mathrm{b}_{ 31 } = 432$, $\mathrm{b}_{ 32 } = 410$, $\mathrm{b}_{ 33 } = 387$, $\mathrm{b}_{ 34 } = 357$, $\mathrm{b}_{ 35 } = 329$, $\mathrm{b}_{ 36 } = 295$, $\mathrm{b}_{ 37 } = 264$, $\mathrm{b}_{ 38 } = 230$, $\mathrm{b}_{ 39 } = 200$, $\mathrm{b}_{ 40 } = 168$, $\mathrm{b}_{ 41 } = 141$, $\mathrm{b}_{ 42 } = 115$, $\mathrm{b}_{ 43 } = 92$, $\mathrm{b}_{ 44 } = 72$, $\mathrm{b}_{ 45 } = 55$, $\mathrm{b}_{ 46 } = 41$, $\mathrm{b}_{ 47 } = 29$, $\mathrm{b}_{ 48 } = 21$, $\mathrm{b}_{ 49 } = 13$, $\mathrm{b}_{ 50 } = 9$, $\mathrm{b}_{ 51 } = 5$, $\mathrm{b}_{ 52 } = 3$, $\mathrm{b}_{ 53 } = 1$, $\mathrm{b}_{ 54 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{4})$
adjoint group of type $\mathrm{E}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{4})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{4})$
133
Projective geometry
minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{4}\hookrightarrow\mathbb{P}^{ 365749 }$

degree
582030940630457808098077820639300812800
Hilbert series
1, 365750, 9183989815, 42331500340584, 59252883172761600, 34076322766338821640, 9815671002827519255250, 1625078000309265938029880, 170864210950800587387041875, 12295167166204266888828545625, 641401493961037390572435787980, 25381253024245583764430441856000, 790045586254351795405044821524480, 19924353584137340978602300332825600, 417159725002297152369930677833997436, 7400344351154142197873770709744159145, 113160899936072585072753606731897711125, 1513532929504221294323717493411468761504, 17930668739932090531010098361947317191860, 190209560832265615573812920907967785839000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{7}/\mathrm{P}_{4})$. Will you be the first to construct one? Let us know if you do!