\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.

Lagrangian Grassmannian $\LGr(6,12)$

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 } &= 1 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 4 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 16 } &= 4 \\ \mathrm{b}_{ 18 } &= 5 \\ \mathrm{b}_{ 20 } &= 5 \\ \mathrm{b}_{ 22 } &= 5 \\ \mathrm{b}_{ 24 } &= 5 \\ \mathrm{b}_{ 26 } &= 4 \\ \mathrm{b}_{ 28 } &= 4 \\ \mathrm{b}_{ 30 } &= 4 \\ \mathrm{b}_{ 32 } &= 3 \\ \mathrm{b}_{ 34 } &= 2 \\ \mathrm{b}_{ 36 } &= 2 \\ \mathrm{b}_{ 38 } &= 1 \\ \mathrm{b}_{ 40 } &= 1 \\ \mathrm{b}_{ 42 } &= 1 \end{align*}
Basic information
dimension
21
index
7
Euler characteristic
64
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 5$, $\mathrm{b}_{ 22 } = 5$, $\mathrm{b}_{ 24 } = 5$, $\mathrm{b}_{ 26 } = 4$, $\mathrm{b}_{ 28 } = 4$, $\mathrm{b}_{ 30 } = 4$, $\mathrm{b}_{ 32 } = 3$, $\mathrm{b}_{ 34 } = 2$, $\mathrm{b}_{ 36 } = 2$, $\mathrm{b}_{ 38 } = 1$, $\mathrm{b}_{ 40 } = 1$, $\mathrm{b}_{ 42 } = 1$
$\mathrm{Aut}^0(\LGr(6,12))$
$\mathrm{PSp}_{ 12 }$
$\pi_0\mathrm{Aut}(\LGr(6,12))$
$1$
$\dim\mathrm{Aut}^0(\LGr(6,12))$
78
Projective geometry
minimal embedding

$\LGr(6,12)\hookrightarrow\mathbb{P}^{ 428 }$

degree
1100742656
Hilbert series
1, 429, 40898, 1643356, 37119160, 553361016, 6018114036, 51067020290, 354544250775, 2085445951875, 10670888978100, 48482555556240, 198799911271104, 745399022720704, 2583249697382640, 8348637902014644, 25350083740894757, 72778756857842321, 198627295537238854, 517731580553810700, ...
Exceptional collections
  • Fonarev constructed a full exceptional sequence in 2019, see arXiv:1911.08968.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\LGr(6,12))$ are given by:

Homological projective duality