\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.

Orthogonal Grassmannian $\OGr(2,14)$

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 } &= 4 \\ \mathrm{b}_{ 12 } &= 5 \\ \mathrm{b}_{ 14 } &= 5 \\ \mathrm{b}_{ 16 } &= 6 \\ \mathrm{b}_{ 18 } &= 6 \\ \mathrm{b}_{ 20 } &= 7 \\ \mathrm{b}_{ 22 } &= 7 \\ \mathrm{b}_{ 24 } &= 6 \\ \mathrm{b}_{ 26 } &= 6 \\ \mathrm{b}_{ 28 } &= 5 \\ \mathrm{b}_{ 30 } &= 5 \\ \mathrm{b}_{ 32 } &= 4 \\ \mathrm{b}_{ 34 } &= 3 \\ \mathrm{b}_{ 36 } &= 2 \\ \mathrm{b}_{ 38 } &= 2 \\ \mathrm{b}_{ 40 } &= 1 \\ \mathrm{b}_{ 42 } &= 1 \end{align*}
Basic information
dimension
21
index
11
Euler characteristic
84
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 4$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 14 } = 5$, $\mathrm{b}_{ 16 } = 6$, $\mathrm{b}_{ 18 } = 6$, $\mathrm{b}_{ 20 } = 7$, $\mathrm{b}_{ 22 } = 7$, $\mathrm{b}_{ 24 } = 6$, $\mathrm{b}_{ 26 } = 6$, $\mathrm{b}_{ 28 } = 5$, $\mathrm{b}_{ 30 } = 5$, $\mathrm{b}_{ 32 } = 4$, $\mathrm{b}_{ 34 } = 3$, $\mathrm{b}_{ 36 } = 2$, $\mathrm{b}_{ 38 } = 2$, $\mathrm{b}_{ 40 } = 1$, $\mathrm{b}_{ 42 } = 1$
$\mathrm{Aut}^0(\OGr(2,14))$
$\mathrm{PSO}_{ 14 }$
$\pi_0\mathrm{Aut}(\OGr(2,14))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\OGr(2,14))$
91
Projective geometry
minimal embedding

$\OGr(2,14)\hookrightarrow\mathbb{P}^{ 90 }$

degree
235144
Hilbert series
1, 91, 3080, 58344, 741741, 7014007, 52676624, 328671200, 1760034276, 8288708428, 34981096384, 134282273216, 474507071220, 1558691535324, 4798442924704, 13938796754960, 38427932768225, 101045111470875, 254498309465400, 616265320671000, ...
Exceptional collections
  • Kuznetsov–Smirnov constructed a full exceptional sequence in 2020, see arXiv:2001.04148.
Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(2,14))$ are given by:

Homological projective duality