# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type E8/P7

Basic information
dimension
83
index
19
Euler characteristic
6720
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 6$, $\mathrm{b}_{ 8 } = 7$, $\mathrm{b}_{ 9 } = 9$, $\mathrm{b}_{ 10 } = 11$, $\mathrm{b}_{ 11 } = 14$, $\mathrm{b}_{ 12 } = 17$, $\mathrm{b}_{ 13 } = 21$, $\mathrm{b}_{ 14 } = 24$, $\mathrm{b}_{ 15 } = 28$, $\mathrm{b}_{ 16 } = 33$, $\mathrm{b}_{ 17 } = 38$, $\mathrm{b}_{ 18 } = 44$, $\mathrm{b}_{ 19 } = 50$, $\mathrm{b}_{ 20 } = 56$, $\mathrm{b}_{ 21 } = 62$, $\mathrm{b}_{ 22 } = 70$, $\mathrm{b}_{ 23 } = 77$, $\mathrm{b}_{ 24 } = 85$, $\mathrm{b}_{ 25 } = 92$, $\mathrm{b}_{ 26 } = 100$, $\mathrm{b}_{ 27 } = 107$, $\mathrm{b}_{ 28 } = 117$, $\mathrm{b}_{ 29 } = 124$, $\mathrm{b}_{ 30 } = 132$, $\mathrm{b}_{ 31 } = 138$, $\mathrm{b}_{ 32 } = 146$, $\mathrm{b}_{ 33 } = 152$, $\mathrm{b}_{ 34 } = 160$, $\mathrm{b}_{ 35 } = 165$, $\mathrm{b}_{ 36 } = 170$, $\mathrm{b}_{ 37 } = 174$, $\mathrm{b}_{ 38 } = 179$, $\mathrm{b}_{ 39 } = 182$, $\mathrm{b}_{ 40 } = 185$, $\mathrm{b}_{ 41 } = 186$, $\mathrm{b}_{ 42 } = 186$, $\mathrm{b}_{ 43 } = 186$, $\mathrm{b}_{ 44 } = 186$, $\mathrm{b}_{ 45 } = 185$, $\mathrm{b}_{ 46 } = 182$, $\mathrm{b}_{ 47 } = 179$, $\mathrm{b}_{ 48 } = 174$, $\mathrm{b}_{ 49 } = 170$, $\mathrm{b}_{ 50 } = 165$, $\mathrm{b}_{ 51 } = 160$, $\mathrm{b}_{ 52 } = 152$, $\mathrm{b}_{ 53 } = 146$, $\mathrm{b}_{ 54 } = 138$, $\mathrm{b}_{ 55 } = 132$, $\mathrm{b}_{ 56 } = 124$, $\mathrm{b}_{ 57 } = 117$, $\mathrm{b}_{ 58 } = 107$, $\mathrm{b}_{ 59 } = 100$, $\mathrm{b}_{ 60 } = 92$, $\mathrm{b}_{ 61 } = 85$, $\mathrm{b}_{ 62 } = 77$, $\mathrm{b}_{ 63 } = 70$, $\mathrm{b}_{ 64 } = 62$, $\mathrm{b}_{ 65 } = 56$, $\mathrm{b}_{ 66 } = 50$, $\mathrm{b}_{ 67 } = 44$, $\mathrm{b}_{ 68 } = 38$, $\mathrm{b}_{ 69 } = 33$, $\mathrm{b}_{ 70 } = 28$, $\mathrm{b}_{ 71 } = 24$, $\mathrm{b}_{ 72 } = 21$, $\mathrm{b}_{ 73 } = 17$, $\mathrm{b}_{ 74 } = 14$, $\mathrm{b}_{ 75 } = 11$, $\mathrm{b}_{ 76 } = 9$, $\mathrm{b}_{ 77 } = 7$, $\mathrm{b}_{ 78 } = 6$, $\mathrm{b}_{ 79 } = 4$, $\mathrm{b}_{ 80 } = 3$, $\mathrm{b}_{ 81 } = 2$, $\mathrm{b}_{ 82 } = 2$, $\mathrm{b}_{ 83 } = 1$, $\mathrm{b}_{ 84 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{7})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{7})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{7})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{7}\hookrightarrow\mathbb{P}^{ 30379 }$

degree
89865581377093987169670453043891751023017984000
Hilbert series
1, 30380, 203205000, 492957660000, 556808824845000, 342155764891414320, 127360452615132304896, 31072820707262801600000, 5277545821530688709458000, 654321479913123595631465000, 61524920198537442851936890080, 4527098239713335928466521786600, 267562846995927541796680726875000, 12984266658899503458138949434984375, 527145171964186971464866423575000000, 18195344962241268750676776882882000000, 541468392947766409236201070761843562500, 14062598230684746894025543740876065625000, 322176120456958557777321515555489451000000, 6573133446781546091748564759915352080000000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{7})$. Will you be the first to construct one? Let us know if you do!