\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/P1

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 5 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 7 } &= 3 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 5 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 11 } &= 7 \\ \mathrm{b}_{ 12 } &= 8 \\ \mathrm{b}_{ 13 } &= 10 \\ \mathrm{b}_{ 14 } &= 11 \\ \mathrm{b}_{ 15 } &= 13 \\ \mathrm{b}_{ 16 } &= 14 \\ \mathrm{b}_{ 17 } &= 17 \\ \mathrm{b}_{ 18 } &= 19 \\ \mathrm{b}_{ 19 } &= 21 \\ \mathrm{b}_{ 20 } &= 23 \\ \mathrm{b}_{ 21 } &= 26 \\ \mathrm{b}_{ 22 } &= 28 \\ \mathrm{b}_{ 23 } &= 31 \\ \mathrm{b}_{ 24 } &= 34 \\ \mathrm{b}_{ 25 } &= 36 \\ \mathrm{b}_{ 26 } &= 38 \\ \mathrm{b}_{ 27 } &= 41 \\ \mathrm{b}_{ 28 } &= 44 \\ \mathrm{b}_{ 29 } &= 46 \\ \mathrm{b}_{ 30 } &= 49 \\ \mathrm{b}_{ 31 } &= 50 \\ \mathrm{b}_{ 32 } &= 52 \\ \mathrm{b}_{ 33 } &= 54 \\ \mathrm{b}_{ 34 } &= 57 \\ \mathrm{b}_{ 35 } &= 57 \\ \mathrm{b}_{ 36 } &= 59 \\ \mathrm{b}_{ 37 } &= 59 \\ \mathrm{b}_{ 38 } &= 60 \\ \mathrm{b}_{ 39 } &= 60 \\ \mathrm{b}_{ 40 } &= 62 \\ \mathrm{b}_{ 41 } &= 60 \\ \mathrm{b}_{ 42 } &= 60 \\ \mathrm{b}_{ 43 } &= 59 \\ \mathrm{b}_{ 44 } &= 59 \\ \mathrm{b}_{ 45 } &= 57 \\ \mathrm{b}_{ 46 } &= 57 \\ \mathrm{b}_{ 47 } &= 54 \\ \mathrm{b}_{ 48 } &= 52 \\ \mathrm{b}_{ 49 } &= 50 \\ \mathrm{b}_{ 50 } &= 49 \\ \mathrm{b}_{ 51 } &= 46 \\ \mathrm{b}_{ 52 } &= 44 \\ \mathrm{b}_{ 53 } &= 41 \\ \mathrm{b}_{ 54 } &= 38 \\ \mathrm{b}_{ 55 } &= 36 \\ \mathrm{b}_{ 56 } &= 34 \\ \mathrm{b}_{ 57 } &= 31 \\ \mathrm{b}_{ 58 } &= 28 \\ \mathrm{b}_{ 59 } &= 26 \\ \mathrm{b}_{ 60 } &= 23 \\ \mathrm{b}_{ 61 } &= 21 \\ \mathrm{b}_{ 62 } &= 19 \\ \mathrm{b}_{ 63 } &= 17 \\ \mathrm{b}_{ 64 } &= 14 \\ \mathrm{b}_{ 65 } &= 13 \\ \mathrm{b}_{ 66 } &= 11 \\ \mathrm{b}_{ 67 } &= 10 \\ \mathrm{b}_{ 68 } &= 8 \\ \mathrm{b}_{ 69 } &= 7 \\ \mathrm{b}_{ 70 } &= 5 \\ \mathrm{b}_{ 71 } &= 5 \\ \mathrm{b}_{ 72 } &= 4 \\ \mathrm{b}_{ 73 } &= 3 \\ \mathrm{b}_{ 74 } &= 2 \\ \mathrm{b}_{ 75 } &= 2 \\ \mathrm{b}_{ 76 } &= 1 \\ \mathrm{b}_{ 77 } &= 1 \\ \mathrm{b}_{ 78 } &= 1 \\ \mathrm{b}_{ 79 } &= 1 \end{align*}
Basic information
dimension
78
index
23
Euler characteristic
2160
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 7 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 5$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 11 } = 7$, $\mathrm{b}_{ 12 } = 8$, $\mathrm{b}_{ 13 } = 10$, $\mathrm{b}_{ 14 } = 11$, $\mathrm{b}_{ 15 } = 13$, $\mathrm{b}_{ 16 } = 14$, $\mathrm{b}_{ 17 } = 17$, $\mathrm{b}_{ 18 } = 19$, $\mathrm{b}_{ 19 } = 21$, $\mathrm{b}_{ 20 } = 23$, $\mathrm{b}_{ 21 } = 26$, $\mathrm{b}_{ 22 } = 28$, $\mathrm{b}_{ 23 } = 31$, $\mathrm{b}_{ 24 } = 34$, $\mathrm{b}_{ 25 } = 36$, $\mathrm{b}_{ 26 } = 38$, $\mathrm{b}_{ 27 } = 41$, $\mathrm{b}_{ 28 } = 44$, $\mathrm{b}_{ 29 } = 46$, $\mathrm{b}_{ 30 } = 49$, $\mathrm{b}_{ 31 } = 50$, $\mathrm{b}_{ 32 } = 52$, $\mathrm{b}_{ 33 } = 54$, $\mathrm{b}_{ 34 } = 57$, $\mathrm{b}_{ 35 } = 57$, $\mathrm{b}_{ 36 } = 59$, $\mathrm{b}_{ 37 } = 59$, $\mathrm{b}_{ 38 } = 60$, $\mathrm{b}_{ 39 } = 60$, $\mathrm{b}_{ 40 } = 62$, $\mathrm{b}_{ 41 } = 60$, $\mathrm{b}_{ 42 } = 60$, $\mathrm{b}_{ 43 } = 59$, $\mathrm{b}_{ 44 } = 59$, $\mathrm{b}_{ 45 } = 57$, $\mathrm{b}_{ 46 } = 57$, $\mathrm{b}_{ 47 } = 54$, $\mathrm{b}_{ 48 } = 52$, $\mathrm{b}_{ 49 } = 50$, $\mathrm{b}_{ 50 } = 49$, $\mathrm{b}_{ 51 } = 46$, $\mathrm{b}_{ 52 } = 44$, $\mathrm{b}_{ 53 } = 41$, $\mathrm{b}_{ 54 } = 38$, $\mathrm{b}_{ 55 } = 36$, $\mathrm{b}_{ 56 } = 34$, $\mathrm{b}_{ 57 } = 31$, $\mathrm{b}_{ 58 } = 28$, $\mathrm{b}_{ 59 } = 26$, $\mathrm{b}_{ 60 } = 23$, $\mathrm{b}_{ 61 } = 21$, $\mathrm{b}_{ 62 } = 19$, $\mathrm{b}_{ 63 } = 17$, $\mathrm{b}_{ 64 } = 14$, $\mathrm{b}_{ 65 } = 13$, $\mathrm{b}_{ 66 } = 11$, $\mathrm{b}_{ 67 } = 10$, $\mathrm{b}_{ 68 } = 8$, $\mathrm{b}_{ 69 } = 7$, $\mathrm{b}_{ 70 } = 5$, $\mathrm{b}_{ 71 } = 5$, $\mathrm{b}_{ 72 } = 4$, $\mathrm{b}_{ 73 } = 3$, $\mathrm{b}_{ 74 } = 2$, $\mathrm{b}_{ 75 } = 2$, $\mathrm{b}_{ 76 } = 1$, $\mathrm{b}_{ 77 } = 1$, $\mathrm{b}_{ 78 } = 1$, $\mathrm{b}_{ 79 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{1})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{1})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{1})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 3874 }$

degree
566737444875521606631975195475968000
Hilbert series
1, 3875, 4881384, 2903770000, 976217435000, 207272853913752, 30016620324194000, 3138280535780660120, 247491761539737346875, 15243441784308510515625, 754291366788666511287000, 30696344926439269703062500, 1047731612488529774828006250, 30497909649841602914892187500, 767999051805484838339634375000, 16939558438056482585955866100000, 330817811768507722440902447250000, 5774915620620045304684694312022000, 90869541917706942429714014772000000, 1298517132972979360163725684963200000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{1})$. Will you be the first to construct one? Let us know if you do!