# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Lagrangian Grassmannian $\LGr(3,6)$

Basic information
dimension
6
index
4
Euler characteristic
8
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 1$
$\mathrm{Aut}^0(\LGr(3,6))$
$\mathrm{PSp}_{ 6 }$
$\pi_0\mathrm{Aut}(\LGr(3,6))$
$1$
$\dim\mathrm{Aut}^0(\LGr(3,6))$
21
Projective geometry
minimal embedding

$\LGr(3,6)\hookrightarrow\mathbb{P}^{ 13 }$

degree
16
Hilbert series
1, 14, 84, 330, 1001, 2548, 5712, 11628, 21945, 38962, 65780, 106470, 166257, 251720, 371008, 534072, 752913, 1041846, 1417780, 1900514, ...
Exceptional collections
• Fonarev constructed a full exceptional sequence in 2019, see arXiv:1911.08968.
• Samokhin constructed a full exceptional sequence in 2001, see MR1859740.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\LGr(3,6))$ are given by:

Homological projective duality