\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 $\SGr(2,14)$

Y Z 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 } &= 6 \\ \mathrm{b}_{ 22 } &= 6 \\ \mathrm{b}_{ 24 } &= 6 \\ \mathrm{b}_{ 26 } &= 6 \\ \mathrm{b}_{ 28 } &= 5 \\ \mathrm{b}_{ 30 } &= 5 \\ \mathrm{b}_{ 32 } &= 4 \\ \mathrm{b}_{ 34 } &= 4 \\ \mathrm{b}_{ 36 } &= 3 \\ \mathrm{b}_{ 38 } &= 3 \\ \mathrm{b}_{ 40 } &= 2 \\ \mathrm{b}_{ 42 } &= 2 \\ \mathrm{b}_{ 44 } &= 1 \\ \mathrm{b}_{ 46 } &= 1 \end{align*}
Basic information
dimension
23
index
13
Euler characteristic
84
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 6$, $\mathrm{b}_{ 22 } = 6$, $\mathrm{b}_{ 24 } = 6$, $\mathrm{b}_{ 26 } = 6$, $\mathrm{b}_{ 28 } = 5$, $\mathrm{b}_{ 30 } = 5$, $\mathrm{b}_{ 32 } = 4$, $\mathrm{b}_{ 34 } = 4$, $\mathrm{b}_{ 36 } = 3$, $\mathrm{b}_{ 38 } = 3$, $\mathrm{b}_{ 40 } = 2$, $\mathrm{b}_{ 42 } = 2$, $\mathrm{b}_{ 44 } = 1$, $\mathrm{b}_{ 46 } = 1$
$\mathrm{Aut}^0(\SGr(2,14))$
$\mathrm{PSp}_{ 14 }$
$\pi_0\mathrm{Aut}(\SGr(2,14))$
$1$
$\dim\mathrm{Aut}^0(\SGr(2,14))$
105
Projective geometry
minimal embedding

$\SGr(2,14)\hookrightarrow\mathbb{P}^{ 89 }$

degree
208012
Hilbert series
1, 90, 3094, 60515, 802620, 7970144, 63117600, 416305656, 2359921980, 11772484360, 52634397816, 213993385060, 800476354664, 2781578066400, 9051166724000, 27766109627100, 80762914630125, 223837027882650, 593640195261750, 1512156523161375, ...
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 $\mathrm{c}_1(\SGr(2,14))$ are given by:

Homological projective duality