# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Symplectic Grassmannian $\SGr(2,6)$

Basic information
dimension
7
index
5
Euler characteristic
12
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 1$, $\mathrm{b}_{ 14 } = 1$
$\mathrm{Aut}^0(\SGr(2,6))$
$\mathrm{PSp}_{ 6 }$
$\pi_0\mathrm{Aut}(\SGr(2,6))$
$1$
$\dim\mathrm{Aut}^0(\SGr(2,6))$
21
Projective geometry
minimal embedding

$\SGr(2,6)\hookrightarrow\mathbb{P}^{ 13 }$

degree
14
Hilbert series
1, 14, 90, 385, 1274, 3528, 8568, 18810, 38115, 72358, 130130, 223587, 369460, 590240, 915552, 1383732, 2043621, 2956590, 4198810, 5863781, ...
Exceptional collections
• Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.
Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\SGr(2,6))$ are given by:

Homological projective duality