\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.

Grassmannian $\Gr(3,14)$

There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 12 } &= 7 \\ \mathrm{b}_{ 14 } &= 8 \\ \mathrm{b}_{ 16 } &= 10 \\ \mathrm{b}_{ 18 } &= 12 \\ \mathrm{b}_{ 20 } &= 14 \\ \mathrm{b}_{ 22 } &= 16 \\ \mathrm{b}_{ 24 } &= 18 \\ \mathrm{b}_{ 26 } &= 19 \\ \mathrm{b}_{ 28 } &= 20 \\ \mathrm{b}_{ 30 } &= 21 \\ \mathrm{b}_{ 32 } &= 21 \\ \mathrm{b}_{ 34 } &= 21 \\ \mathrm{b}_{ 36 } &= 21 \\ \mathrm{b}_{ 38 } &= 20 \\ \mathrm{b}_{ 40 } &= 19 \\ \mathrm{b}_{ 42 } &= 18 \\ \mathrm{b}_{ 44 } &= 16 \\ \mathrm{b}_{ 46 } &= 14 \\ \mathrm{b}_{ 48 } &= 12 \\ \mathrm{b}_{ 50 } &= 10 \\ \mathrm{b}_{ 52 } &= 8 \\ \mathrm{b}_{ 54 } &= 7 \\ \mathrm{b}_{ 56 } &= 5 \\ \mathrm{b}_{ 58 } &= 4 \\ \mathrm{b}_{ 60 } &= 3 \\ \mathrm{b}_{ 62 } &= 2 \\ \mathrm{b}_{ 64 } &= 1 \\ \mathrm{b}_{ 66 } &= 1 \end{align*}
Basic information
dimension
33
index
14
Euler characteristic
364
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 10$, $\mathrm{b}_{ 18 } = 12$, $\mathrm{b}_{ 20 } = 14$, $\mathrm{b}_{ 22 } = 16$, $\mathrm{b}_{ 24 } = 18$, $\mathrm{b}_{ 26 } = 19$, $\mathrm{b}_{ 28 } = 20$, $\mathrm{b}_{ 30 } = 21$, $\mathrm{b}_{ 32 } = 21$, $\mathrm{b}_{ 34 } = 21$, $\mathrm{b}_{ 36 } = 21$, $\mathrm{b}_{ 38 } = 20$, $\mathrm{b}_{ 40 } = 19$, $\mathrm{b}_{ 42 } = 18$, $\mathrm{b}_{ 44 } = 16$, $\mathrm{b}_{ 46 } = 14$, $\mathrm{b}_{ 48 } = 12$, $\mathrm{b}_{ 50 } = 10$, $\mathrm{b}_{ 52 } = 8$, $\mathrm{b}_{ 54 } = 7$, $\mathrm{b}_{ 56 } = 5$, $\mathrm{b}_{ 58 } = 4$, $\mathrm{b}_{ 60 } = 3$, $\mathrm{b}_{ 62 } = 2$, $\mathrm{b}_{ 64 } = 1$, $\mathrm{b}_{ 66 } = 1$
$\mathrm{Aut}^0(\Gr(3,14))$
$\mathrm{PGL}_{ 14 }$
$\pi_0\mathrm{Aut}(\Gr(3,14))$
$1$
$\dim\mathrm{Aut}^0(\Gr(3,14))$
195
Projective geometry
minimal embedding

$\Gr(3,14)\hookrightarrow\mathbb{P}^{ 363 }$

degree
145862174640
Hilbert series
1, 364, 41405, 2318680, 78835120, 1837984512, 31803696288, 431621592480, 4783805983320, 44648855844320, 359423289546776, 2543610972177184, 16072267131888800, 91841526467936000, 479707973069130000, 2311063705550691000, 10348807328532138375, 43360327919340022500, 170969012278450351875, 637598722482040410000, ...
Exceptional collections
  • Kapranov constructed a full exceptional sequence in 1988, see MR0939472.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

Homological projective duality