\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 $\LGr(6,12)$

Betti numbers
\begin{align*} \mathrm{b}_{ 1 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 3 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 5 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 7 } &= 4 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 9 } &= 4 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 11 } &= 5 \\ \mathrm{b}_{ 12 } &= 5 \\ \mathrm{b}_{ 13 } &= 5 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 15 } &= 4 \\ \mathrm{b}_{ 16 } &= 4 \\ \mathrm{b}_{ 17 } &= 3 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 19 } &= 2 \\ \mathrm{b}_{ 20 } &= 1 \\ \mathrm{b}_{ 21 } &= 1 \\ \mathrm{b}_{ 22 } &= 1 \end{align*}
Basic information
dimension
21
index
7
Euler characteristic
64
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 4$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 9 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 11 } = 5$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 13 } = 5$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 15 } = 4$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 17 } = 3$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 19 } = 2$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 21 } = 1$, $\mathrm{b}_{ 22 } = 1$
$\mathrm{Aut}^0(\LGr(6,12))$
$\mathrm{PSp}_{ 12 }$
$\pi_0\mathrm{Aut}(\LGr(6,12))$
$1$
$\dim\mathrm{Aut}^0(\LGr(6,12))$
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.