\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.

Symplectic Grassmannian $\SGr(3,8)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 2 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 5 } &= 3 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 7 } &= 4 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 11 } &= 2 \\ \mathrm{b}_{ 12 } &= 1 \\ \mathrm{b}_{ 13 } &= 1 \end{align*}
Basic information
dimension
12
index
6
Euler characteristic
32
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 2$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 5 } = 3$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 11 } = 2$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 13 } = 1$
$\mathrm{Aut}^0(\SGr(3,8))$
$\mathrm{PSp}_{ 8 }$
$\pi_0\mathrm{Aut}(\SGr(3,8))$
$1$
$\dim\mathrm{Aut}^0(\SGr(3,8))$
36
Projective geometry
minimal embedding

$\SGr(3,8)\hookrightarrow\mathbb{P}^{ 47 }$

degree
2112
Hilbert series
1, 48, 825, 8008, 53508, 274176, 1151172, 4138200, 13132977, 37626160, 98963865, 242070192, 556301200, 1210970112, 2513648016, 5002784208, 9590713353, 17778961200, 31975197793, 55950522936, ...
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