\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 $\LGr(7,14)$

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 } &= 1 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 4 \\ \mathrm{b}_{ 14 } &= 5 \\ \mathrm{b}_{ 16 } &= 5 \\ \mathrm{b}_{ 18 } &= 6 \\ \mathrm{b}_{ 20 } &= 7 \\ \mathrm{b}_{ 22 } &= 7 \\ \mathrm{b}_{ 24 } &= 8 \\ \mathrm{b}_{ 26 } &= 8 \\ \mathrm{b}_{ 28 } &= 8 \\ \mathrm{b}_{ 30 } &= 8 \\ \mathrm{b}_{ 32 } &= 8 \\ \mathrm{b}_{ 34 } &= 7 \\ \mathrm{b}_{ 36 } &= 7 \\ \mathrm{b}_{ 38 } &= 6 \\ \mathrm{b}_{ 40 } &= 5 \\ \mathrm{b}_{ 42 } &= 5 \\ \mathrm{b}_{ 44 } &= 4 \\ \mathrm{b}_{ 46 } &= 3 \\ \mathrm{b}_{ 48 } &= 2 \\ \mathrm{b}_{ 50 } &= 2 \\ \mathrm{b}_{ 52 } &= 1 \\ \mathrm{b}_{ 54 } &= 1 \\ \mathrm{b}_{ 56 } &= 1 \end{align*}
Basic information
dimension
28
index
8
Euler characteristic
128
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 5$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 6$, $\mathrm{b}_{ 20 } = 7$, $\mathrm{b}_{ 22 } = 7$, $\mathrm{b}_{ 24 } = 8$, $\mathrm{b}_{ 26 } = 8$, $\mathrm{b}_{ 28 } = 8$, $\mathrm{b}_{ 30 } = 8$, $\mathrm{b}_{ 32 } = 8$, $\mathrm{b}_{ 34 } = 7$, $\mathrm{b}_{ 36 } = 7$, $\mathrm{b}_{ 38 } = 6$, $\mathrm{b}_{ 40 } = 5$, $\mathrm{b}_{ 42 } = 5$, $\mathrm{b}_{ 44 } = 4$, $\mathrm{b}_{ 46 } = 3$, $\mathrm{b}_{ 48 } = 2$, $\mathrm{b}_{ 50 } = 2$, $\mathrm{b}_{ 52 } = 1$, $\mathrm{b}_{ 54 } = 1$, $\mathrm{b}_{ 56 } = 1$
$\mathrm{Aut}^0(\LGr(7,14))$
$\mathrm{PSp}_{ 14 }$
$\pi_0\mathrm{Aut}(\LGr(7,14))$
$1$
$\dim\mathrm{Aut}^0(\LGr(7,14))$
105
Projective geometry
minimal embedding

$\LGr(7,14)\hookrightarrow\mathbb{P}^{ 1429 }$

degree
48608795688960
Hilbert series
1, 1430, 379236, 37119160, 1844536720, 55804330152, 1153471856900, 17618122000050, 210309203300625, 2045253720802500, 16726481666902800, 117924492822177600, 731033150646450432, 4049210782061424320, 20306456189361772560, 93218107934594843356, 395354056003669808305, 1561373056164500587866, 5780818251270332339300, 20181961451776258981000, ...
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 $\mathrm{c}_1(\LGr(7,14))$ are given by:

Homological projective duality