\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/P8

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 6 } &= 1 \\ \mathrm{b}_{ 8 } &= 1 \\ \mathrm{b}_{ 10 } &= 1 \\ \mathrm{b}_{ 12 } &= 2 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 16 } &= 2 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 20 } &= 3 \\ \mathrm{b}_{ 22 } &= 3 \\ \mathrm{b}_{ 24 } &= 4 \\ \mathrm{b}_{ 26 } &= 4 \\ \mathrm{b}_{ 28 } &= 4 \\ \mathrm{b}_{ 30 } &= 4 \\ \mathrm{b}_{ 32 } &= 5 \\ \mathrm{b}_{ 34 } &= 5 \\ \mathrm{b}_{ 36 } &= 6 \\ \mathrm{b}_{ 38 } &= 6 \\ \mathrm{b}_{ 40 } &= 6 \\ \mathrm{b}_{ 42 } &= 6 \\ \mathrm{b}_{ 44 } &= 7 \\ \mathrm{b}_{ 46 } &= 7 \\ \mathrm{b}_{ 48 } &= 7 \\ \mathrm{b}_{ 50 } &= 7 \\ \mathrm{b}_{ 52 } &= 7 \\ \mathrm{b}_{ 54 } &= 7 \\ \mathrm{b}_{ 56 } &= 8 \\ \mathrm{b}_{ 58 } &= 8 \\ \mathrm{b}_{ 60 } &= 7 \\ \mathrm{b}_{ 62 } &= 7 \\ \mathrm{b}_{ 64 } &= 7 \\ \mathrm{b}_{ 66 } &= 7 \\ \mathrm{b}_{ 68 } &= 7 \\ \mathrm{b}_{ 70 } &= 7 \\ \mathrm{b}_{ 72 } &= 6 \\ \mathrm{b}_{ 74 } &= 6 \\ \mathrm{b}_{ 76 } &= 6 \\ \mathrm{b}_{ 78 } &= 6 \\ \mathrm{b}_{ 80 } &= 5 \\ \mathrm{b}_{ 82 } &= 5 \\ \mathrm{b}_{ 84 } &= 4 \\ \mathrm{b}_{ 86 } &= 4 \\ \mathrm{b}_{ 88 } &= 4 \\ \mathrm{b}_{ 90 } &= 4 \\ \mathrm{b}_{ 92 } &= 3 \\ \mathrm{b}_{ 94 } &= 3 \\ \mathrm{b}_{ 96 } &= 2 \\ \mathrm{b}_{ 98 } &= 2 \\ \mathrm{b}_{ 100 } &= 2 \\ \mathrm{b}_{ 102 } &= 2 \\ \mathrm{b}_{ 104 } &= 1 \\ \mathrm{b}_{ 106 } &= 1 \\ \mathrm{b}_{ 108 } &= 1 \\ \mathrm{b}_{ 110 } &= 1 \\ \mathrm{b}_{ 112 } &= 1 \\ \mathrm{b}_{ 114 } &= 1 \end{align*}
Basic information
dimension
57
index
29
Euler characteristic
240
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 3$, $\mathrm{b}_{ 22 } = 3$, $\mathrm{b}_{ 24 } = 4$, $\mathrm{b}_{ 26 } = 4$, $\mathrm{b}_{ 28 } = 4$, $\mathrm{b}_{ 30 } = 4$, $\mathrm{b}_{ 32 } = 5$, $\mathrm{b}_{ 34 } = 5$, $\mathrm{b}_{ 36 } = 6$, $\mathrm{b}_{ 38 } = 6$, $\mathrm{b}_{ 40 } = 6$, $\mathrm{b}_{ 42 } = 6$, $\mathrm{b}_{ 44 } = 7$, $\mathrm{b}_{ 46 } = 7$, $\mathrm{b}_{ 48 } = 7$, $\mathrm{b}_{ 50 } = 7$, $\mathrm{b}_{ 52 } = 7$, $\mathrm{b}_{ 54 } = 7$, $\mathrm{b}_{ 56 } = 8$, $\mathrm{b}_{ 58 } = 8$, $\mathrm{b}_{ 60 } = 7$, $\mathrm{b}_{ 62 } = 7$, $\mathrm{b}_{ 64 } = 7$, $\mathrm{b}_{ 66 } = 7$, $\mathrm{b}_{ 68 } = 7$, $\mathrm{b}_{ 70 } = 7$, $\mathrm{b}_{ 72 } = 6$, $\mathrm{b}_{ 74 } = 6$, $\mathrm{b}_{ 76 } = 6$, $\mathrm{b}_{ 78 } = 6$, $\mathrm{b}_{ 80 } = 5$, $\mathrm{b}_{ 82 } = 5$, $\mathrm{b}_{ 84 } = 4$, $\mathrm{b}_{ 86 } = 4$, $\mathrm{b}_{ 88 } = 4$, $\mathrm{b}_{ 90 } = 4$, $\mathrm{b}_{ 92 } = 3$, $\mathrm{b}_{ 94 } = 3$, $\mathrm{b}_{ 96 } = 2$, $\mathrm{b}_{ 98 } = 2$, $\mathrm{b}_{ 100 } = 2$, $\mathrm{b}_{ 102 } = 2$, $\mathrm{b}_{ 104 } = 1$, $\mathrm{b}_{ 106 } = 1$, $\mathrm{b}_{ 108 } = 1$, $\mathrm{b}_{ 110 } = 1$, $\mathrm{b}_{ 112 } = 1$, $\mathrm{b}_{ 114 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{8})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{8})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{8})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{8}\hookrightarrow\mathbb{P}^{ 247 }$

degree
126937516885200
Hilbert series
1, 248, 27000, 1763125, 79143000, 2642777280, 69176971200, 1473701482500, 26284473168750, 401283501480000, 5338265882241600, 62790857238950100, 661062273763905000, 6294003651511200000, 54675736068345120000, 436687003868825311200, 3228153165040477279320, 22217485351372039512000, 143102432756681687640000, 866595309136135835343000, ...
Exceptional collections

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

Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{8}/\mathrm{P}_{8})$ are given by:

Homological projective duality