\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 $\LGr(5,10)$

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 } &= 3 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 9 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 11 } &= 3 \\ \mathrm{b}_{ 12 } &= 2 \\ \mathrm{b}_{ 13 } &= 2 \\ \mathrm{b}_{ 14 } &= 1 \\ \mathrm{b}_{ 15 } &= 1 \\ \mathrm{b}_{ 16 } &= 1 \end{align*}
Basic information
dimension
15
index
6
Euler characteristic
32
Betti numbers
$\mathrm{b}_{ 1 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 3 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 5 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 7 } = 3$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 9 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 11 } = 3$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 13 } = 2$, $\mathrm{b}_{ 14 } = 1$, $\mathrm{b}_{ 15 } = 1$, $\mathrm{b}_{ 16 } = 1$
$\mathrm{Aut}^0(\LGr(5,10))$
$\mathrm{PSp}_{ 10 }$
$\pi_0\mathrm{Aut}(\LGr(5,10))$
$1$
$\dim\mathrm{Aut}^0(\LGr(5,10))$
55
Projective geometry
minimal embedding

$\LGr(5,10)\hookrightarrow\mathbb{P}^{ 131 }$

degree
292864
Hilbert series
1, 132, 4719, 81796, 884884, 6852768, 41314284, 204951252, 869562265, 3245256300, 10880587575, 33309352440, 94307358288, 249485071616, 621856804272, 1470540624696, 3318218562009, 7179339254516, 14955909351383, 30104651175324, ...
Exceptional collections
  • Fonarev constructed a full exceptional sequence in 2019, see arXiv:1911.08968.