# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type E7/P2

Basic information
dimension
42
index
14
Euler characteristic
576
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 5$, $\mathrm{b}_{ 8 } = 7$, $\mathrm{b}_{ 9 } = 8$, $\mathrm{b}_{ 10 } = 10$, $\mathrm{b}_{ 11 } = 12$, $\mathrm{b}_{ 12 } = 14$, $\mathrm{b}_{ 13 } = 16$, $\mathrm{b}_{ 14 } = 18$, $\mathrm{b}_{ 15 } = 20$, $\mathrm{b}_{ 16 } = 22$, $\mathrm{b}_{ 17 } = 24$, $\mathrm{b}_{ 18 } = 25$, $\mathrm{b}_{ 19 } = 26$, $\mathrm{b}_{ 20 } = 27$, $\mathrm{b}_{ 21 } = 28$, $\mathrm{b}_{ 22 } = 28$, $\mathrm{b}_{ 23 } = 28$, $\mathrm{b}_{ 24 } = 27$, $\mathrm{b}_{ 25 } = 26$, $\mathrm{b}_{ 26 } = 25$, $\mathrm{b}_{ 27 } = 24$, $\mathrm{b}_{ 28 } = 22$, $\mathrm{b}_{ 29 } = 20$, $\mathrm{b}_{ 30 } = 18$, $\mathrm{b}_{ 31 } = 16$, $\mathrm{b}_{ 32 } = 14$, $\mathrm{b}_{ 33 } = 12$, $\mathrm{b}_{ 34 } = 10$, $\mathrm{b}_{ 35 } = 8$, $\mathrm{b}_{ 36 } = 7$, $\mathrm{b}_{ 37 } = 5$, $\mathrm{b}_{ 38 } = 4$, $\mathrm{b}_{ 39 } = 3$, $\mathrm{b}_{ 40 } = 2$, $\mathrm{b}_{ 41 } = 1$, $\mathrm{b}_{ 42 } = 1$, $\mathrm{b}_{ 43 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{2})$
adjoint group of type $\mathrm{E}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{2})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{2})$
133
Projective geometry
minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 911 }$

degree
230629602357872640
Hilbert series
1, 912, 253935, 32995248, 2457458575, 118861039440, 4061118864660, 104114789500800, 2095350182853408, 34279106505180160, 468620731551643872, 5474059247498898816, 55645606128266708700, 499788275805501361360, 4017247956270188974125, 29213221999317397602000, 193993785997407232711875, 1185918539273268050250000, 6720932040786435334078125, 35529086969201421823500000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{7}/\mathrm{P}_{2})$. Will you be the first to construct one? Let us know if you do!