# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type E7/P4

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!