\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 E8/P8

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 5 } &= 1 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 7 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 9 } &= 2 \\ \mathrm{b}_{ 10 } &= 2 \\ \mathrm{b}_{ 11 } &= 3 \\ \mathrm{b}_{ 12 } &= 3 \\ \mathrm{b}_{ 13 } &= 4 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 15 } &= 4 \\ \mathrm{b}_{ 16 } &= 4 \\ \mathrm{b}_{ 17 } &= 5 \\ \mathrm{b}_{ 18 } &= 5 \\ \mathrm{b}_{ 19 } &= 6 \\ \mathrm{b}_{ 20 } &= 6 \\ \mathrm{b}_{ 21 } &= 6 \\ \mathrm{b}_{ 22 } &= 6 \\ \mathrm{b}_{ 23 } &= 7 \\ \mathrm{b}_{ 24 } &= 7 \\ \mathrm{b}_{ 25 } &= 7 \\ \mathrm{b}_{ 26 } &= 7 \\ \mathrm{b}_{ 27 } &= 7 \\ \mathrm{b}_{ 28 } &= 7 \\ \mathrm{b}_{ 29 } &= 8 \\ \mathrm{b}_{ 30 } &= 8 \\ \mathrm{b}_{ 31 } &= 7 \\ \mathrm{b}_{ 32 } &= 7 \\ \mathrm{b}_{ 33 } &= 7 \\ \mathrm{b}_{ 34 } &= 7 \\ \mathrm{b}_{ 35 } &= 7 \\ \mathrm{b}_{ 36 } &= 7 \\ \mathrm{b}_{ 37 } &= 6 \\ \mathrm{b}_{ 38 } &= 6 \\ \mathrm{b}_{ 39 } &= 6 \\ \mathrm{b}_{ 40 } &= 6 \\ \mathrm{b}_{ 41 } &= 5 \\ \mathrm{b}_{ 42 } &= 5 \\ \mathrm{b}_{ 43 } &= 4 \\ \mathrm{b}_{ 44 } &= 4 \\ \mathrm{b}_{ 45 } &= 4 \\ \mathrm{b}_{ 46 } &= 4 \\ \mathrm{b}_{ 47 } &= 3 \\ \mathrm{b}_{ 48 } &= 3 \\ \mathrm{b}_{ 49 } &= 2 \\ \mathrm{b}_{ 50 } &= 2 \\ \mathrm{b}_{ 51 } &= 2 \\ \mathrm{b}_{ 52 } &= 2 \\ \mathrm{b}_{ 53 } &= 1 \\ \mathrm{b}_{ 54 } &= 1 \\ \mathrm{b}_{ 55 } &= 1 \\ \mathrm{b}_{ 56 } &= 1 \\ \mathrm{b}_{ 57 } &= 1 \\ \mathrm{b}_{ 58 } &= 1 \end{align*}
Basic information
dimension
57
index
29
Euler characteristic
240
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 7 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 9 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 11 } = 3$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 13 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 15 } = 4$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 17 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 19 } = 6$, $\mathrm{b}_{ 20 } = 6$, $\mathrm{b}_{ 21 } = 6$, $\mathrm{b}_{ 22 } = 6$, $\mathrm{b}_{ 23 } = 7$, $\mathrm{b}_{ 24 } = 7$, $\mathrm{b}_{ 25 } = 7$, $\mathrm{b}_{ 26 } = 7$, $\mathrm{b}_{ 27 } = 7$, $\mathrm{b}_{ 28 } = 7$, $\mathrm{b}_{ 29 } = 8$, $\mathrm{b}_{ 30 } = 8$, $\mathrm{b}_{ 31 } = 7$, $\mathrm{b}_{ 32 } = 7$, $\mathrm{b}_{ 33 } = 7$, $\mathrm{b}_{ 34 } = 7$, $\mathrm{b}_{ 35 } = 7$, $\mathrm{b}_{ 36 } = 7$, $\mathrm{b}_{ 37 } = 6$, $\mathrm{b}_{ 38 } = 6$, $\mathrm{b}_{ 39 } = 6$, $\mathrm{b}_{ 40 } = 6$, $\mathrm{b}_{ 41 } = 5$, $\mathrm{b}_{ 42 } = 5$, $\mathrm{b}_{ 43 } = 4$, $\mathrm{b}_{ 44 } = 4$, $\mathrm{b}_{ 45 } = 4$, $\mathrm{b}_{ 46 } = 4$, $\mathrm{b}_{ 47 } = 3$, $\mathrm{b}_{ 48 } = 3$, $\mathrm{b}_{ 49 } = 2$, $\mathrm{b}_{ 50 } = 2$, $\mathrm{b}_{ 51 } = 2$, $\mathrm{b}_{ 52 } = 2$, $\mathrm{b}_{ 53 } = 1$, $\mathrm{b}_{ 54 } = 1$, $\mathrm{b}_{ 55 } = 1$, $\mathrm{b}_{ 56 } = 1$, $\mathrm{b}_{ 57 } = 1$, $\mathrm{b}_{ 58 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{8})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{8})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{8})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{8}\hookrightarrow\mathbb{P}^{ 247 }$

degree
126937516885200
Hilbert series
1, 248, 27000, 1763125, 79143000, 2642777280, 69176971200, 1473701482500, 26284473168750, 401283501480000, 5338265882241600, 62790857238950100, 661062273763905000, 6294003651511200000, 54675736068345120000, 436687003868825311200, 3228153165040477279320, 22217485351372039512000, 143102432756681687640000, 866595309136135835343000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{8})$. Will you be the first to construct one? Let us know if you do!