\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/P7

Betti numbers
\begin{align*} \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 \end{align*}
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!