\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/P5

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 5 \\ \mathrm{b}_{ 10 } &= 6 \\ \mathrm{b}_{ 12 } &= 8 \\ \mathrm{b}_{ 14 } &= 10 \\ \mathrm{b}_{ 16 } &= 12 \\ \mathrm{b}_{ 18 } &= 13 \\ \mathrm{b}_{ 20 } &= 15 \\ \mathrm{b}_{ 22 } &= 16 \\ \mathrm{b}_{ 24 } &= 16 \\ \mathrm{b}_{ 26 } &= 16 \\ \mathrm{b}_{ 28 } &= 16 \\ \mathrm{b}_{ 30 } &= 15 \\ \mathrm{b}_{ 32 } &= 13 \\ \mathrm{b}_{ 34 } &= 12 \\ \mathrm{b}_{ 36 } &= 10 \\ \mathrm{b}_{ 38 } &= 8 \\ \mathrm{b}_{ 40 } &= 6 \\ \mathrm{b}_{ 42 } &= 5 \\ \mathrm{b}_{ 44 } &= 3 \\ \mathrm{b}_{ 46 } &= 2 \\ \mathrm{b}_{ 48 } &= 1 \\ \mathrm{b}_{ 50 } &= 1 \end{align*}
Basic information
dimension
25
index
9
Euler characteristic
216
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 12 } = 8$, $\mathrm{b}_{ 14 } = 10$, $\mathrm{b}_{ 16 } = 12$, $\mathrm{b}_{ 18 } = 13$, $\mathrm{b}_{ 20 } = 15$, $\mathrm{b}_{ 22 } = 16$, $\mathrm{b}_{ 24 } = 16$, $\mathrm{b}_{ 26 } = 16$, $\mathrm{b}_{ 28 } = 16$, $\mathrm{b}_{ 30 } = 15$, $\mathrm{b}_{ 32 } = 13$, $\mathrm{b}_{ 34 } = 12$, $\mathrm{b}_{ 36 } = 10$, $\mathrm{b}_{ 38 } = 8$, $\mathrm{b}_{ 40 } = 6$, $\mathrm{b}_{ 42 } = 5$, $\mathrm{b}_{ 44 } = 3$, $\mathrm{b}_{ 46 } = 2$, $\mathrm{b}_{ 48 } = 1$, $\mathrm{b}_{ 50 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{6}/\mathrm{P}_{5})$
adjoint group of type $\mathrm{E}_{ 6 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{6}/\mathrm{P}_{5})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{6}/\mathrm{P}_{5})$
78
Projective geometry
minimal embedding

$\mathrm{E}_{6}/\mathrm{P}_{5}\hookrightarrow\mathbb{P}^{ 350 }$

degree
6119118720
Hilbert series
1, 351, 34398, 1559376, 41442192, 740779182, 9731535957, 99876290085, 836979819390, 5919754500660, 36248385923568, 196056157459968, 951851777730048, 4202954077177584, 17062684674908652, 64266967368492174, 226311318775941399, 749969516734275915, 2352031814236869050, 7014911034217086000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{6}/\mathrm{P}_{5})$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{6}/\mathrm{P}_{5})$ are given by:

Homological projective duality