\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 $\Gr(5,8)$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 5 } &= 4 \\ \mathrm{b}_{ 6 } &= 5 \\ \mathrm{b}_{ 7 } &= 6 \\ \mathrm{b}_{ 8 } &= 6 \\ \mathrm{b}_{ 9 } &= 6 \\ \mathrm{b}_{ 10 } &= 6 \\ \mathrm{b}_{ 11 } &= 5 \\ \mathrm{b}_{ 12 } &= 4 \\ \mathrm{b}_{ 13 } &= 3 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 15 } &= 1 \\ \mathrm{b}_{ 16 } &= 1 \end{align*}
Basic information
dimension
15
index
8
Euler characteristic
56
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 4$, $\mathrm{b}_{ 6 } = 5$, $\mathrm{b}_{ 7 } = 6$, $\mathrm{b}_{ 8 } = 6$, $\mathrm{b}_{ 9 } = 6$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 11 } = 5$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 13 } = 3$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 15 } = 1$, $\mathrm{b}_{ 16 } = 1$
$\mathrm{Aut}^0(\Gr(5,8))$
$\mathrm{PGL}_{ 8 }$
$\pi_0\mathrm{Aut}(\Gr(5,8))$
$1$
$\dim\mathrm{Aut}^0(\Gr(5,8))$
63
Projective geometry
minimal embedding

$\Gr(5,8)\hookrightarrow\mathbb{P}^{ 55 }$

degree
6006
Hilbert series
1, 56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480, 1178943365600, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.