\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}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 4 \\ \mathrm{b}_{ 5 } &= 6 \\ \mathrm{b}_{ 6 } &= 9 \\ \mathrm{b}_{ 7 } &= 13 \\ \mathrm{b}_{ 8 } &= 18 \\ \mathrm{b}_{ 9 } &= 24 \\ \mathrm{b}_{ 10 } &= 32 \\ \mathrm{b}_{ 11 } &= 40 \\ \mathrm{b}_{ 12 } &= 50 \\ \mathrm{b}_{ 13 } &= 61 \\ \mathrm{b}_{ 14 } &= 73 \\ \mathrm{b}_{ 15 } &= 85 \\ \mathrm{b}_{ 16 } &= 99 \\ \mathrm{b}_{ 17 } &= 112 \\ \mathrm{b}_{ 18 } &= 125 \\ \mathrm{b}_{ 19 } &= 138 \\ \mathrm{b}_{ 20 } &= 150 \\ \mathrm{b}_{ 21 } &= 161 \\ \mathrm{b}_{ 22 } &= 170 \\ \mathrm{b}_{ 23 } &= 178 \\ \mathrm{b}_{ 24 } &= 183 \\ \mathrm{b}_{ 25 } &= 187 \\ \mathrm{b}_{ 26 } &= 188 \\ \mathrm{b}_{ 27 } &= 187 \\ \mathrm{b}_{ 28 } &= 183 \\ \mathrm{b}_{ 29 } &= 178 \\ \mathrm{b}_{ 30 } &= 170 \\ \mathrm{b}_{ 31 } &= 161 \\ \mathrm{b}_{ 32 } &= 150 \\ \mathrm{b}_{ 33 } &= 138 \\ \mathrm{b}_{ 34 } &= 125 \\ \mathrm{b}_{ 35 } &= 112 \\ \mathrm{b}_{ 36 } &= 99 \\ \mathrm{b}_{ 37 } &= 85 \\ \mathrm{b}_{ 38 } &= 73 \\ \mathrm{b}_{ 39 } &= 61 \\ \mathrm{b}_{ 40 } &= 50 \\ \mathrm{b}_{ 41 } &= 40 \\ \mathrm{b}_{ 42 } &= 32 \\ \mathrm{b}_{ 43 } &= 24 \\ \mathrm{b}_{ 44 } &= 18 \\ \mathrm{b}_{ 45 } &= 13 \\ \mathrm{b}_{ 46 } &= 9 \\ \mathrm{b}_{ 47 } &= 6 \\ \mathrm{b}_{ 48 } &= 4 \\ \mathrm{b}_{ 49 } &= 2 \\ \mathrm{b}_{ 50 } &= 1 \\ \mathrm{b}_{ 51 } &= 1 \end{align*}
Basic information
dimension
50
index
10
Euler characteristic
4032
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 4$, $\mathrm{b}_{ 5 } = 6$, $\mathrm{b}_{ 6 } = 9$, $\mathrm{b}_{ 7 } = 13$, $\mathrm{b}_{ 8 } = 18$, $\mathrm{b}_{ 9 } = 24$, $\mathrm{b}_{ 10 } = 32$, $\mathrm{b}_{ 11 } = 40$, $\mathrm{b}_{ 12 } = 50$, $\mathrm{b}_{ 13 } = 61$, $\mathrm{b}_{ 14 } = 73$, $\mathrm{b}_{ 15 } = 85$, $\mathrm{b}_{ 16 } = 99$, $\mathrm{b}_{ 17 } = 112$, $\mathrm{b}_{ 18 } = 125$, $\mathrm{b}_{ 19 } = 138$, $\mathrm{b}_{ 20 } = 150$, $\mathrm{b}_{ 21 } = 161$, $\mathrm{b}_{ 22 } = 170$, $\mathrm{b}_{ 23 } = 178$, $\mathrm{b}_{ 24 } = 183$, $\mathrm{b}_{ 25 } = 187$, $\mathrm{b}_{ 26 } = 188$, $\mathrm{b}_{ 27 } = 187$, $\mathrm{b}_{ 28 } = 183$, $\mathrm{b}_{ 29 } = 178$, $\mathrm{b}_{ 30 } = 170$, $\mathrm{b}_{ 31 } = 161$, $\mathrm{b}_{ 32 } = 150$, $\mathrm{b}_{ 33 } = 138$, $\mathrm{b}_{ 34 } = 125$, $\mathrm{b}_{ 35 } = 112$, $\mathrm{b}_{ 36 } = 99$, $\mathrm{b}_{ 37 } = 85$, $\mathrm{b}_{ 38 } = 73$, $\mathrm{b}_{ 39 } = 61$, $\mathrm{b}_{ 40 } = 50$, $\mathrm{b}_{ 41 } = 40$, $\mathrm{b}_{ 42 } = 32$, $\mathrm{b}_{ 43 } = 24$, $\mathrm{b}_{ 44 } = 18$, $\mathrm{b}_{ 45 } = 13$, $\mathrm{b}_{ 46 } = 9$, $\mathrm{b}_{ 47 } = 6$, $\mathrm{b}_{ 48 } = 4$, $\mathrm{b}_{ 49 } = 2$, $\mathrm{b}_{ 50 } = 1$, $\mathrm{b}_{ 51 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{5})$
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!