\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 $\SGr(2,12)$

Y Z Y Z 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 } &= 4 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 16 } &= 5 \\ \mathrm{b}_{ 18 } &= 5 \\ \mathrm{b}_{ 20 } &= 5 \\ \mathrm{b}_{ 22 } &= 5 \\ \mathrm{b}_{ 24 } &= 4 \\ \mathrm{b}_{ 26 } &= 4 \\ \mathrm{b}_{ 28 } &= 3 \\ \mathrm{b}_{ 30 } &= 3 \\ \mathrm{b}_{ 32 } &= 2 \\ \mathrm{b}_{ 34 } &= 2 \\ \mathrm{b}_{ 36 } &= 1 \\ \mathrm{b}_{ 38 } &= 1 \end{align*}
Basic information
dimension
19
index
11
Euler characteristic
60
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 5$, $\mathrm{b}_{ 22 } = 5$, $\mathrm{b}_{ 24 } = 4$, $\mathrm{b}_{ 26 } = 4$, $\mathrm{b}_{ 28 } = 3$, $\mathrm{b}_{ 30 } = 3$, $\mathrm{b}_{ 32 } = 2$, $\mathrm{b}_{ 34 } = 2$, $\mathrm{b}_{ 36 } = 1$, $\mathrm{b}_{ 38 } = 1$
$\mathrm{Aut}^0(\SGr(2,12))$
$\mathrm{PSp}_{ 12 }$
$\pi_0\mathrm{Aut}(\SGr(2,12))$
$1$
$\dim\mathrm{Aut}^0(\SGr(2,12))$
78
Projective geometry
minimal embedding

$\SGr(2,12)\hookrightarrow\mathbb{P}^{ 64 }$

degree
16796
Hilbert series
1, 65, 1650, 24310, 247247, 1912911, 11971960, 63206000, 290115540, 1184101204, 4372637048, 14810544840, 46520301100, 136727327660, 378824441424, 995628339640, 2495320309625, 5990816885625, 13831429862250, 30813266033550, ...
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 $\mathrm{c}_1(\SGr(2,12))$ are given by:

Homological projective duality