\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 $\SGr(2,8)$

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 } &= 2 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 3 \\ \mathrm{b}_{ 14 } &= 3 \\ \mathrm{b}_{ 16 } &= 2 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 20 } &= 1 \\ \mathrm{b}_{ 22 } &= 1 \end{align*}
Basic information
dimension
11
index
7
Euler characteristic
24
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 14 } = 3$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 22 } = 1$
$\mathrm{Aut}^0(\SGr(2,8))$
$\mathrm{PSp}_{ 8 }$
$\pi_0\mathrm{Aut}(\SGr(2,8))$
$1$
$\dim\mathrm{Aut}^0(\SGr(2,8))$
36
Projective geometry
minimal embedding

$\SGr(2,8)\hookrightarrow\mathbb{P}^{ 26 }$

degree
132
Hilbert series
1, 27, 308, 2184, 11340, 47124, 165528, 509652, 1410981, 3578575, 8432424, 18663008, 39132912, 78279696, 150233760, 277932456, 497594097, 865014843, 1464269884, 2419540200, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.
Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\SGr(2,8))$ are given by:

Homological projective duality