\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 E6/P4

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 6 } &= 5 \\ \mathrm{b}_{ 8 } &= 8 \\ \mathrm{b}_{ 10 } &= 11 \\ \mathrm{b}_{ 12 } &= 17 \\ \mathrm{b}_{ 14 } &= 21 \\ \mathrm{b}_{ 16 } &= 27 \\ \mathrm{b}_{ 18 } &= 33 \\ \mathrm{b}_{ 20 } &= 39 \\ \mathrm{b}_{ 22 } &= 43 \\ \mathrm{b}_{ 24 } &= 48 \\ \mathrm{b}_{ 26 } &= 51 \\ \mathrm{b}_{ 28 } &= 52 \\ \mathrm{b}_{ 30 } &= 52 \\ \mathrm{b}_{ 32 } &= 51 \\ \mathrm{b}_{ 34 } &= 48 \\ \mathrm{b}_{ 36 } &= 43 \\ \mathrm{b}_{ 38 } &= 39 \\ \mathrm{b}_{ 40 } &= 33 \\ \mathrm{b}_{ 42 } &= 27 \\ \mathrm{b}_{ 44 } &= 21 \\ \mathrm{b}_{ 46 } &= 17 \\ \mathrm{b}_{ 48 } &= 11 \\ \mathrm{b}_{ 50 } &= 8 \\ \mathrm{b}_{ 52 } &= 5 \\ \mathrm{b}_{ 54 } &= 3 \\ \mathrm{b}_{ 56 } &= 1 \\ \mathrm{b}_{ 58 } &= 1 \end{align*}
Basic information
dimension
29
index
7
Euler characteristic
720
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 6 } = 5$, $\mathrm{b}_{ 8 } = 8$, $\mathrm{b}_{ 10 } = 11$, $\mathrm{b}_{ 12 } = 17$, $\mathrm{b}_{ 14 } = 21$, $\mathrm{b}_{ 16 } = 27$, $\mathrm{b}_{ 18 } = 33$, $\mathrm{b}_{ 20 } = 39$, $\mathrm{b}_{ 22 } = 43$, $\mathrm{b}_{ 24 } = 48$, $\mathrm{b}_{ 26 } = 51$, $\mathrm{b}_{ 28 } = 52$, $\mathrm{b}_{ 30 } = 52$, $\mathrm{b}_{ 32 } = 51$, $\mathrm{b}_{ 34 } = 48$, $\mathrm{b}_{ 36 } = 43$, $\mathrm{b}_{ 38 } = 39$, $\mathrm{b}_{ 40 } = 33$, $\mathrm{b}_{ 42 } = 27$, $\mathrm{b}_{ 44 } = 21$, $\mathrm{b}_{ 46 } = 17$, $\mathrm{b}_{ 48 } = 11$, $\mathrm{b}_{ 50 } = 8$, $\mathrm{b}_{ 52 } = 5$, $\mathrm{b}_{ 54 } = 3$, $\mathrm{b}_{ 56 } = 1$, $\mathrm{b}_{ 58 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{6}/\mathrm{P}_{4})$
adjoint group of type $\mathrm{E}_{ 6 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{6}/\mathrm{P}_{4})$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\mathrm{E}_{6}/\mathrm{P}_{4})$
78
Projective geometry
minimal embedding

$\mathrm{E}_{6}/\mathrm{P}_{4}\hookrightarrow\mathbb{P}^{ 2924 }$

degree
6976089058498560
Hilbert series
1, 2925, 1337050, 200449886, 14059334016, 566175692160, 14913879339150, 280792241343900, 4025210543883815, 46036602501049875, 435295075917928590, 3498456896925135600, 24431989764207001600, 150929317381761515520, 836896418801677703664, 4216266165250964998134, 19497202099068251870325, 83475895135311127273825, 333355743987571380619544, 1249635948403762700988570, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{6}/\mathrm{P}_{4})$. 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 not known yet to be generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{6}/\mathrm{P}_{4})$ are given by:

Homological projective duality