\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 $\SGr(3,8)$

Y Z Y Z Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 10 } &= 4 \\ \mathrm{b}_{ 12 } &= 4 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 16 } &= 3 \\ \mathrm{b}_{ 18 } &= 3 \\ \mathrm{b}_{ 20 } &= 2 \\ \mathrm{b}_{ 22 } &= 1 \\ \mathrm{b}_{ 24 } &= 1 \end{align*}
Basic information
dimension
12
index
6
Euler characteristic
32
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 4$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 3$, $\mathrm{b}_{ 18 } = 3$, $\mathrm{b}_{ 20 } = 2$, $\mathrm{b}_{ 22 } = 1$, $\mathrm{b}_{ 24 } = 1$
$\mathrm{Aut}^0(\SGr(3,8))$
$\mathrm{PSp}_{ 8 }$
$\pi_0\mathrm{Aut}(\SGr(3,8))$
$1$
$\dim\mathrm{Aut}^0(\SGr(3,8))$
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, ...
Exceptional collections
Quantum cohomology

The small quantum cohomology is not known to be (non-)semisimple.

The big quantum cohomology is not known yet to be generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\SGr(3,8))$ are given by:

Homological projective duality