\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 E7/P6

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 12 } &= 7 \\ \mathrm{b}_{ 14 } &= 8 \\ \mathrm{b}_{ 16 } &= 11 \\ \mathrm{b}_{ 18 } &= 13 \\ \mathrm{b}_{ 20 } &= 16 \\ \mathrm{b}_{ 22 } &= 18 \\ \mathrm{b}_{ 24 } &= 21 \\ \mathrm{b}_{ 26 } &= 24 \\ \mathrm{b}_{ 28 } &= 26 \\ \mathrm{b}_{ 30 } &= 29 \\ \mathrm{b}_{ 32 } &= 31 \\ \mathrm{b}_{ 34 } &= 34 \\ \mathrm{b}_{ 36 } &= 34 \\ \mathrm{b}_{ 38 } &= 36 \\ \mathrm{b}_{ 40 } &= 36 \\ \mathrm{b}_{ 42 } &= 38 \\ \mathrm{b}_{ 44 } &= 36 \\ \mathrm{b}_{ 46 } &= 36 \\ \mathrm{b}_{ 48 } &= 34 \\ \mathrm{b}_{ 50 } &= 34 \\ \mathrm{b}_{ 52 } &= 31 \\ \mathrm{b}_{ 54 } &= 29 \\ \mathrm{b}_{ 56 } &= 26 \\ \mathrm{b}_{ 58 } &= 24 \\ \mathrm{b}_{ 60 } &= 21 \\ \mathrm{b}_{ 62 } &= 18 \\ \mathrm{b}_{ 64 } &= 16 \\ \mathrm{b}_{ 66 } &= 13 \\ \mathrm{b}_{ 68 } &= 11 \\ \mathrm{b}_{ 70 } &= 8 \\ \mathrm{b}_{ 72 } &= 7 \\ \mathrm{b}_{ 74 } &= 5 \\ \mathrm{b}_{ 76 } &= 4 \\ \mathrm{b}_{ 78 } &= 2 \\ \mathrm{b}_{ 80 } &= 2 \\ \mathrm{b}_{ 82 } &= 1 \\ \mathrm{b}_{ 84 } &= 1 \end{align*}
Basic information
dimension
42
index
13
Euler characteristic
756
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 11$, $\mathrm{b}_{ 18 } = 13$, $\mathrm{b}_{ 20 } = 16$, $\mathrm{b}_{ 22 } = 18$, $\mathrm{b}_{ 24 } = 21$, $\mathrm{b}_{ 26 } = 24$, $\mathrm{b}_{ 28 } = 26$, $\mathrm{b}_{ 30 } = 29$, $\mathrm{b}_{ 32 } = 31$, $\mathrm{b}_{ 34 } = 34$, $\mathrm{b}_{ 36 } = 34$, $\mathrm{b}_{ 38 } = 36$, $\mathrm{b}_{ 40 } = 36$, $\mathrm{b}_{ 42 } = 38$, $\mathrm{b}_{ 44 } = 36$, $\mathrm{b}_{ 46 } = 36$, $\mathrm{b}_{ 48 } = 34$, $\mathrm{b}_{ 50 } = 34$, $\mathrm{b}_{ 52 } = 31$, $\mathrm{b}_{ 54 } = 29$, $\mathrm{b}_{ 56 } = 26$, $\mathrm{b}_{ 58 } = 24$, $\mathrm{b}_{ 60 } = 21$, $\mathrm{b}_{ 62 } = 18$, $\mathrm{b}_{ 64 } = 16$, $\mathrm{b}_{ 66 } = 13$, $\mathrm{b}_{ 68 } = 11$, $\mathrm{b}_{ 70 } = 8$, $\mathrm{b}_{ 72 } = 7$, $\mathrm{b}_{ 74 } = 5$, $\mathrm{b}_{ 76 } = 4$, $\mathrm{b}_{ 78 } = 2$, $\mathrm{b}_{ 80 } = 2$, $\mathrm{b}_{ 82 } = 1$, $\mathrm{b}_{ 84 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{6})$
adjoint group of type $\mathrm{E}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{6})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{6})$
133
Projective geometry
minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{6}\hookrightarrow\mathbb{P}^{ 1538 }$

degree
6457628866020433920
Hilbert series
1, 1539, 617253, 105489615, 9743909175, 561104814270, 22155294211050, 641798777779380, 14341812253735200, 256924158460700640, 3803151504372077088, 47661484991340720864, 515805481647912249276, 4900345026363722587050, 41434653012551675362750, 315469600749446851604325, 2184413433392140346347875, 13874772955031952078740625, 81446133440497985177484375, 444734992730941305619078125, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{7}/\mathrm{P}_{6})$. 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 $\mathrm{c}_1(\mathrm{E}_{7}/\mathrm{P}_{6})$ are given by:

Homological projective duality