# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type E8/P8

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!