\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/P5

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 8 } &= 6 \\ \mathrm{b}_{ 10 } &= 9 \\ \mathrm{b}_{ 12 } &= 13 \\ \mathrm{b}_{ 14 } &= 18 \\ \mathrm{b}_{ 16 } &= 24 \\ \mathrm{b}_{ 18 } &= 32 \\ \mathrm{b}_{ 20 } &= 40 \\ \mathrm{b}_{ 22 } &= 50 \\ \mathrm{b}_{ 24 } &= 61 \\ \mathrm{b}_{ 26 } &= 73 \\ \mathrm{b}_{ 28 } &= 85 \\ \mathrm{b}_{ 30 } &= 99 \\ \mathrm{b}_{ 32 } &= 112 \\ \mathrm{b}_{ 34 } &= 125 \\ \mathrm{b}_{ 36 } &= 138 \\ \mathrm{b}_{ 38 } &= 150 \\ \mathrm{b}_{ 40 } &= 161 \\ \mathrm{b}_{ 42 } &= 170 \\ \mathrm{b}_{ 44 } &= 178 \\ \mathrm{b}_{ 46 } &= 183 \\ \mathrm{b}_{ 48 } &= 187 \\ \mathrm{b}_{ 50 } &= 188 \\ \mathrm{b}_{ 52 } &= 187 \\ \mathrm{b}_{ 54 } &= 183 \\ \mathrm{b}_{ 56 } &= 178 \\ \mathrm{b}_{ 58 } &= 170 \\ \mathrm{b}_{ 60 } &= 161 \\ \mathrm{b}_{ 62 } &= 150 \\ \mathrm{b}_{ 64 } &= 138 \\ \mathrm{b}_{ 66 } &= 125 \\ \mathrm{b}_{ 68 } &= 112 \\ \mathrm{b}_{ 70 } &= 99 \\ \mathrm{b}_{ 72 } &= 85 \\ \mathrm{b}_{ 74 } &= 73 \\ \mathrm{b}_{ 76 } &= 61 \\ \mathrm{b}_{ 78 } &= 50 \\ \mathrm{b}_{ 80 } &= 40 \\ \mathrm{b}_{ 82 } &= 32 \\ \mathrm{b}_{ 84 } &= 24 \\ \mathrm{b}_{ 86 } &= 18 \\ \mathrm{b}_{ 88 } &= 13 \\ \mathrm{b}_{ 90 } &= 9 \\ \mathrm{b}_{ 92 } &= 6 \\ \mathrm{b}_{ 94 } &= 4 \\ \mathrm{b}_{ 96 } &= 2 \\ \mathrm{b}_{ 98 } &= 1 \\ \mathrm{b}_{ 100 } &= 1 \end{align*}
Basic information
dimension
50
index
10
Euler characteristic
4032
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 8 } = 6$, $\mathrm{b}_{ 10 } = 9$, $\mathrm{b}_{ 12 } = 13$, $\mathrm{b}_{ 14 } = 18$, $\mathrm{b}_{ 16 } = 24$, $\mathrm{b}_{ 18 } = 32$, $\mathrm{b}_{ 20 } = 40$, $\mathrm{b}_{ 22 } = 50$, $\mathrm{b}_{ 24 } = 61$, $\mathrm{b}_{ 26 } = 73$, $\mathrm{b}_{ 28 } = 85$, $\mathrm{b}_{ 30 } = 99$, $\mathrm{b}_{ 32 } = 112$, $\mathrm{b}_{ 34 } = 125$, $\mathrm{b}_{ 36 } = 138$, $\mathrm{b}_{ 38 } = 150$, $\mathrm{b}_{ 40 } = 161$, $\mathrm{b}_{ 42 } = 170$, $\mathrm{b}_{ 44 } = 178$, $\mathrm{b}_{ 46 } = 183$, $\mathrm{b}_{ 48 } = 187$, $\mathrm{b}_{ 50 } = 188$, $\mathrm{b}_{ 52 } = 187$, $\mathrm{b}_{ 54 } = 183$, $\mathrm{b}_{ 56 } = 178$, $\mathrm{b}_{ 58 } = 170$, $\mathrm{b}_{ 60 } = 161$, $\mathrm{b}_{ 62 } = 150$, $\mathrm{b}_{ 64 } = 138$, $\mathrm{b}_{ 66 } = 125$, $\mathrm{b}_{ 68 } = 112$, $\mathrm{b}_{ 70 } = 99$, $\mathrm{b}_{ 72 } = 85$, $\mathrm{b}_{ 74 } = 73$, $\mathrm{b}_{ 76 } = 61$, $\mathrm{b}_{ 78 } = 50$, $\mathrm{b}_{ 80 } = 40$, $\mathrm{b}_{ 82 } = 32$, $\mathrm{b}_{ 84 } = 24$, $\mathrm{b}_{ 86 } = 18$, $\mathrm{b}_{ 88 } = 13$, $\mathrm{b}_{ 90 } = 9$, $\mathrm{b}_{ 92 } = 6$, $\mathrm{b}_{ 94 } = 4$, $\mathrm{b}_{ 96 } = 2$, $\mathrm{b}_{ 98 } = 1$, $\mathrm{b}_{ 100 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{5})$
adjoint group of type $\mathrm{E}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{5})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{5})$
133
Projective geometry
minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{5}\hookrightarrow\mathbb{P}^{ 27663 }$

degree
7189094617369512864411864268800
Hilbert series
1, 27664, 109120648, 122211210240, 55316934276120, 12558300006045024, 1652084830739933590, 139491450390045039000, 8150914397472701604750, 349067632878637409842560, 11457227739469333306727424, 298660275350729176734760960, 6363906479714806736751313920, 113502798476673004420339183488, 1728171187811885553111503151894, 22838477706875454786731446453576, 265685045394342480356361068851540, 2753789609279788667104987042619200, 25697281478091630912981145307715000, 217858949465800585414942950480000000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{7}/\mathrm{P}_{5})$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology

The small quantum cohomology is not known to be (non-)semisimple.

The big quantum cohomology is not known yet to be generically semisimple.

Homological projective duality