\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.

Grassmannian $\Gr(4,8)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 5 } &= 5 \\ \mathrm{b}_{ 6 } &= 5 \\ \mathrm{b}_{ 7 } &= 7 \\ \mathrm{b}_{ 8 } &= 7 \\ \mathrm{b}_{ 9 } &= 8 \\ \mathrm{b}_{ 10 } &= 7 \\ \mathrm{b}_{ 11 } &= 7 \\ \mathrm{b}_{ 12 } &= 5 \\ \mathrm{b}_{ 13 } &= 5 \\ \mathrm{b}_{ 14 } &= 3 \\ \mathrm{b}_{ 15 } &= 2 \\ \mathrm{b}_{ 16 } &= 1 \\ \mathrm{b}_{ 17 } &= 1 \end{align*}
Basic information
dimension
16
index
8
Euler characteristic
70
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 5$, $\mathrm{b}_{ 6 } = 5$, $\mathrm{b}_{ 7 } = 7$, $\mathrm{b}_{ 8 } = 7$, $\mathrm{b}_{ 9 } = 8$, $\mathrm{b}_{ 10 } = 7$, $\mathrm{b}_{ 11 } = 7$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 13 } = 5$, $\mathrm{b}_{ 14 } = 3$, $\mathrm{b}_{ 15 } = 2$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 17 } = 1$
$\mathrm{Aut}^0(\Gr(4,8))$
$\mathrm{PGL}_{ 8 }$
$\pi_0\mathrm{Aut}(\Gr(4,8))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\Gr(4,8))$
63
Projective geometry
minimal embedding

$\Gr(4,8)\hookrightarrow\mathbb{P}^{ 69 }$

degree
24024
Hilbert series
1, 70, 1764, 24696, 232848, 1646568, 9343620, 44537922, 184225041, 677352676, 2254684432, 6892441920, 19571505408, 52101067968, 131018862096, 313203587004, 715536058545, 1569305708586, 3316911815140, 6778924352200, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.