# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type E8/P1

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!