\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.

Orthogonal Grassmannian $\OGr(4,17)$

Y Z 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 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 5 \\ \mathrm{b}_{ 10 } &= 6 \\ \mathrm{b}_{ 12 } &= 9 \\ \mathrm{b}_{ 14 } &= 11 \\ \mathrm{b}_{ 16 } &= 15 \\ \mathrm{b}_{ 18 } &= 18 \\ \mathrm{b}_{ 20 } &= 22 \\ \mathrm{b}_{ 22 } &= 26 \\ \mathrm{b}_{ 24 } &= 31 \\ \mathrm{b}_{ 26 } &= 35 \\ \mathrm{b}_{ 28 } &= 39 \\ \mathrm{b}_{ 30 } &= 44 \\ \mathrm{b}_{ 32 } &= 47 \\ \mathrm{b}_{ 34 } &= 51 \\ \mathrm{b}_{ 36 } &= 53 \\ \mathrm{b}_{ 38 } &= 56 \\ \mathrm{b}_{ 40 } &= 56 \\ \mathrm{b}_{ 42 } &= 58 \\ \mathrm{b}_{ 44 } &= 56 \\ \mathrm{b}_{ 46 } &= 56 \\ \mathrm{b}_{ 48 } &= 53 \\ \mathrm{b}_{ 50 } &= 51 \\ \mathrm{b}_{ 52 } &= 47 \\ \mathrm{b}_{ 54 } &= 44 \\ \mathrm{b}_{ 56 } &= 39 \\ \mathrm{b}_{ 58 } &= 35 \\ \mathrm{b}_{ 60 } &= 31 \\ \mathrm{b}_{ 62 } &= 26 \\ \mathrm{b}_{ 64 } &= 22 \\ \mathrm{b}_{ 66 } &= 18 \\ \mathrm{b}_{ 68 } &= 15 \\ \mathrm{b}_{ 70 } &= 11 \\ \mathrm{b}_{ 72 } &= 9 \\ \mathrm{b}_{ 74 } &= 6 \\ \mathrm{b}_{ 76 } &= 5 \\ \mathrm{b}_{ 78 } &= 3 \\ \mathrm{b}_{ 80 } &= 2 \\ \mathrm{b}_{ 82 } &= 1 \\ \mathrm{b}_{ 84 } &= 1 \end{align*}
Basic information
dimension
42
index
12
Euler characteristic
1120
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 12 } = 9$, $\mathrm{b}_{ 14 } = 11$, $\mathrm{b}_{ 16 } = 15$, $\mathrm{b}_{ 18 } = 18$, $\mathrm{b}_{ 20 } = 22$, $\mathrm{b}_{ 22 } = 26$, $\mathrm{b}_{ 24 } = 31$, $\mathrm{b}_{ 26 } = 35$, $\mathrm{b}_{ 28 } = 39$, $\mathrm{b}_{ 30 } = 44$, $\mathrm{b}_{ 32 } = 47$, $\mathrm{b}_{ 34 } = 51$, $\mathrm{b}_{ 36 } = 53$, $\mathrm{b}_{ 38 } = 56$, $\mathrm{b}_{ 40 } = 56$, $\mathrm{b}_{ 42 } = 58$, $\mathrm{b}_{ 44 } = 56$, $\mathrm{b}_{ 46 } = 56$, $\mathrm{b}_{ 48 } = 53$, $\mathrm{b}_{ 50 } = 51$, $\mathrm{b}_{ 52 } = 47$, $\mathrm{b}_{ 54 } = 44$, $\mathrm{b}_{ 56 } = 39$, $\mathrm{b}_{ 58 } = 35$, $\mathrm{b}_{ 60 } = 31$, $\mathrm{b}_{ 62 } = 26$, $\mathrm{b}_{ 64 } = 22$, $\mathrm{b}_{ 66 } = 18$, $\mathrm{b}_{ 68 } = 15$, $\mathrm{b}_{ 70 } = 11$, $\mathrm{b}_{ 72 } = 9$, $\mathrm{b}_{ 74 } = 6$, $\mathrm{b}_{ 76 } = 5$, $\mathrm{b}_{ 78 } = 3$, $\mathrm{b}_{ 80 } = 2$, $\mathrm{b}_{ 82 } = 1$, $\mathrm{b}_{ 84 } = 1$
$\mathrm{Aut}^0(\OGr(4,17))$
$\mathrm{SO}_{ 17 }$
$\pi_0\mathrm{Aut}(\OGr(4,17))$
$1$
$\dim\mathrm{Aut}^0(\OGr(4,17))$
136
Projective geometry
minimal embedding

$\OGr(4,17)\hookrightarrow\mathbb{P}^{ 2379 }$

degree
146372920963129835520
Hilbert series
1, 2380, 1317840, 289199988, 32638486881, 2215644651220, 100408229922000, 3270269401542000, 80842531177633680, 1581394412076475776, 25291124779521922048, 339444408158712474240, 3905462501554712046480, 39201216605241330944976, 348352428236434448867920, 2774671833684335263224700, 20020186201407204044120625, 132049563296166164611462500, 802492042265713634684850000, 4524453856754071139973937500, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
28774013606441248
self-dual
no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Tevelev, Theorems 7.1–7.2 (arXiv version)] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\OGr(4,17))$:

Quantum cohomology
small
semisimplicity unknown
big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Homological projective duality

No Lefschetz decomposition recorded.

No HP-dual model recorded.

$k$-Fanoness

Fano but not 2-Fano. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Carolina Araujo; Roya Beheshti; Ana-Maria Castravet; Kelly Jabbusch; Svetlana Makarova; Enrica Mazzon; Libby Taylor; Nivedita Viswanathan. Higher Fano Manifolds. Rev. Un. Mat. Argentina 64 (2022), 103–125. arXiv:2110.02339.
  2. Alexander Samokhin; Wilberd van der Kallen. Highest weight category structures on $\operatorname{rep}(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$. Preprint (2024). arXiv:2407.13653.
  3. Changzheng Li; Zhaoyang Liu; Tony Yue Yu; Chi Zhang. Convergence and semisimplicity of big quantum cohomology of flag varieties. Preprint (2026). arXiv:2610.28418.

All references.