\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/P2

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 3 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 7 } &= 6 \\ \mathrm{b}_{ 8 } &= 8 \\ \mathrm{b}_{ 9 } &= 10 \\ \mathrm{b}_{ 10 } &= 13 \\ \mathrm{b}_{ 11 } &= 17 \\ \mathrm{b}_{ 12 } &= 21 \\ \mathrm{b}_{ 13 } &= 26 \\ \mathrm{b}_{ 14 } &= 32 \\ \mathrm{b}_{ 15 } &= 38 \\ \mathrm{b}_{ 16 } &= 46 \\ \mathrm{b}_{ 17 } &= 55 \\ \mathrm{b}_{ 18 } &= 64 \\ \mathrm{b}_{ 19 } &= 74 \\ \mathrm{b}_{ 20 } &= 86 \\ \mathrm{b}_{ 21 } &= 98 \\ \mathrm{b}_{ 22 } &= 112 \\ \mathrm{b}_{ 23 } &= 127 \\ \mathrm{b}_{ 24 } &= 142 \\ \mathrm{b}_{ 25 } &= 157 \\ \mathrm{b}_{ 26 } &= 175 \\ \mathrm{b}_{ 27 } &= 193 \\ \mathrm{b}_{ 28 } &= 211 \\ \mathrm{b}_{ 29 } &= 230 \\ \mathrm{b}_{ 30 } &= 249 \\ \mathrm{b}_{ 31 } &= 267 \\ \mathrm{b}_{ 32 } &= 287 \\ \mathrm{b}_{ 33 } &= 307 \\ \mathrm{b}_{ 34 } &= 325 \\ \mathrm{b}_{ 35 } &= 343 \\ \mathrm{b}_{ 36 } &= 361 \\ \mathrm{b}_{ 37 } &= 377 \\ \mathrm{b}_{ 38 } &= 393 \\ \mathrm{b}_{ 39 } &= 409 \\ \mathrm{b}_{ 40 } &= 421 \\ \mathrm{b}_{ 41 } &= 432 \\ \mathrm{b}_{ 42 } &= 443 \\ \mathrm{b}_{ 43 } &= 452 \\ \mathrm{b}_{ 44 } &= 458 \\ \mathrm{b}_{ 45 } &= 464 \\ \mathrm{b}_{ 46 } &= 466 \\ \mathrm{b}_{ 47 } &= 466 \\ \mathrm{b}_{ 48 } &= 466 \\ \mathrm{b}_{ 49 } &= 464 \\ \mathrm{b}_{ 50 } &= 458 \\ \mathrm{b}_{ 51 } &= 452 \\ \mathrm{b}_{ 52 } &= 443 \\ \mathrm{b}_{ 53 } &= 432 \\ \mathrm{b}_{ 54 } &= 421 \\ \mathrm{b}_{ 55 } &= 409 \\ \mathrm{b}_{ 56 } &= 393 \\ \mathrm{b}_{ 57 } &= 377 \\ \mathrm{b}_{ 58 } &= 361 \\ \mathrm{b}_{ 59 } &= 343 \\ \mathrm{b}_{ 60 } &= 325 \\ \mathrm{b}_{ 61 } &= 307 \\ \mathrm{b}_{ 62 } &= 287 \\ \mathrm{b}_{ 63 } &= 267 \\ \mathrm{b}_{ 64 } &= 249 \\ \mathrm{b}_{ 65 } &= 230 \\ \mathrm{b}_{ 66 } &= 211 \\ \mathrm{b}_{ 67 } &= 193 \\ \mathrm{b}_{ 68 } &= 175 \\ \mathrm{b}_{ 69 } &= 157 \\ \mathrm{b}_{ 70 } &= 142 \\ \mathrm{b}_{ 71 } &= 127 \\ \mathrm{b}_{ 72 } &= 112 \\ \mathrm{b}_{ 73 } &= 98 \\ \mathrm{b}_{ 74 } &= 86 \\ \mathrm{b}_{ 75 } &= 74 \\ \mathrm{b}_{ 76 } &= 64 \\ \mathrm{b}_{ 77 } &= 55 \\ \mathrm{b}_{ 78 } &= 46 \\ \mathrm{b}_{ 79 } &= 38 \\ \mathrm{b}_{ 80 } &= 32 \\ \mathrm{b}_{ 81 } &= 26 \\ \mathrm{b}_{ 82 } &= 21 \\ \mathrm{b}_{ 83 } &= 17 \\ \mathrm{b}_{ 84 } &= 13 \\ \mathrm{b}_{ 85 } &= 10 \\ \mathrm{b}_{ 86 } &= 8 \\ \mathrm{b}_{ 87 } &= 6 \\ \mathrm{b}_{ 88 } &= 4 \\ \mathrm{b}_{ 89 } &= 3 \\ \mathrm{b}_{ 90 } &= 2 \\ \mathrm{b}_{ 91 } &= 1 \\ \mathrm{b}_{ 92 } &= 1 \\ \mathrm{b}_{ 93 } &= 1 \end{align*}
Basic information
dimension
92
index
17
Euler characteristic
17280
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 6$, $\mathrm{b}_{ 8 } = 8$, $\mathrm{b}_{ 9 } = 10$, $\mathrm{b}_{ 10 } = 13$, $\mathrm{b}_{ 11 } = 17$, $\mathrm{b}_{ 12 } = 21$, $\mathrm{b}_{ 13 } = 26$, $\mathrm{b}_{ 14 } = 32$, $\mathrm{b}_{ 15 } = 38$, $\mathrm{b}_{ 16 } = 46$, $\mathrm{b}_{ 17 } = 55$, $\mathrm{b}_{ 18 } = 64$, $\mathrm{b}_{ 19 } = 74$, $\mathrm{b}_{ 20 } = 86$, $\mathrm{b}_{ 21 } = 98$, $\mathrm{b}_{ 22 } = 112$, $\mathrm{b}_{ 23 } = 127$, $\mathrm{b}_{ 24 } = 142$, $\mathrm{b}_{ 25 } = 157$, $\mathrm{b}_{ 26 } = 175$, $\mathrm{b}_{ 27 } = 193$, $\mathrm{b}_{ 28 } = 211$, $\mathrm{b}_{ 29 } = 230$, $\mathrm{b}_{ 30 } = 249$, $\mathrm{b}_{ 31 } = 267$, $\mathrm{b}_{ 32 } = 287$, $\mathrm{b}_{ 33 } = 307$, $\mathrm{b}_{ 34 } = 325$, $\mathrm{b}_{ 35 } = 343$, $\mathrm{b}_{ 36 } = 361$, $\mathrm{b}_{ 37 } = 377$, $\mathrm{b}_{ 38 } = 393$, $\mathrm{b}_{ 39 } = 409$, $\mathrm{b}_{ 40 } = 421$, $\mathrm{b}_{ 41 } = 432$, $\mathrm{b}_{ 42 } = 443$, $\mathrm{b}_{ 43 } = 452$, $\mathrm{b}_{ 44 } = 458$, $\mathrm{b}_{ 45 } = 464$, $\mathrm{b}_{ 46 } = 466$, $\mathrm{b}_{ 47 } = 466$, $\mathrm{b}_{ 48 } = 466$, $\mathrm{b}_{ 49 } = 464$, $\mathrm{b}_{ 50 } = 458$, $\mathrm{b}_{ 51 } = 452$, $\mathrm{b}_{ 52 } = 443$, $\mathrm{b}_{ 53 } = 432$, $\mathrm{b}_{ 54 } = 421$, $\mathrm{b}_{ 55 } = 409$, $\mathrm{b}_{ 56 } = 393$, $\mathrm{b}_{ 57 } = 377$, $\mathrm{b}_{ 58 } = 361$, $\mathrm{b}_{ 59 } = 343$, $\mathrm{b}_{ 60 } = 325$, $\mathrm{b}_{ 61 } = 307$, $\mathrm{b}_{ 62 } = 287$, $\mathrm{b}_{ 63 } = 267$, $\mathrm{b}_{ 64 } = 249$, $\mathrm{b}_{ 65 } = 230$, $\mathrm{b}_{ 66 } = 211$, $\mathrm{b}_{ 67 } = 193$, $\mathrm{b}_{ 68 } = 175$, $\mathrm{b}_{ 69 } = 157$, $\mathrm{b}_{ 70 } = 142$, $\mathrm{b}_{ 71 } = 127$, $\mathrm{b}_{ 72 } = 112$, $\mathrm{b}_{ 73 } = 98$, $\mathrm{b}_{ 74 } = 86$, $\mathrm{b}_{ 75 } = 74$, $\mathrm{b}_{ 76 } = 64$, $\mathrm{b}_{ 77 } = 55$, $\mathrm{b}_{ 78 } = 46$, $\mathrm{b}_{ 79 } = 38$, $\mathrm{b}_{ 80 } = 32$, $\mathrm{b}_{ 81 } = 26$, $\mathrm{b}_{ 82 } = 21$, $\mathrm{b}_{ 83 } = 17$, $\mathrm{b}_{ 84 } = 13$, $\mathrm{b}_{ 85 } = 10$, $\mathrm{b}_{ 86 } = 8$, $\mathrm{b}_{ 87 } = 6$, $\mathrm{b}_{ 88 } = 4$, $\mathrm{b}_{ 89 } = 3$, $\mathrm{b}_{ 90 } = 2$, $\mathrm{b}_{ 91 } = 1$, $\mathrm{b}_{ 92 } = 1$, $\mathrm{b}_{ 93 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{2})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{2})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{2})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 147249 }$

degree
221742396458192785546942875610662293134814379227347943424000
Hilbert series
1, 147250, 4076399250, 35361935272950, 125982160881890625, 219535295360061727800, 211712396994333259128000, 123868068014021389904832000, 47200473064255798138436218368, 12390986693062787651882950840000, 2345118041783809242861398688412888, 332154157389659136763618228106828625, 36320209923251219664609477939100007500, 3147687810989929917787834478573310805500, 221097506373270548223283132912819251562500, 12831460733037551669320439047215561510234375, 625611363932726266078933184258231580175781250, 26000628460280646817980180919271683490332031250, 932950905359274517791652426815027754329677812500, 29229869285264902491764188998651481144840509375000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{2})$. Will you be the first to construct one? Let us know if you do!