\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.

Generalised Grassmannian of type F4/P4

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 } &= 1 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 2 \\ \mathrm{b}_{ 12 } &= 2 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 16 } &= 2 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 20 } &= 2 \\ \mathrm{b}_{ 22 } &= 2 \\ \mathrm{b}_{ 24 } &= 1 \\ \mathrm{b}_{ 26 } &= 1 \\ \mathrm{b}_{ 28 } &= 1 \\ \mathrm{b}_{ 30 } &= 1 \end{align*}
Basic information
dimension
15
index
11
Euler characteristic
24
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 2$, $\mathrm{b}_{ 22 } = 2$, $\mathrm{b}_{ 24 } = 1$, $\mathrm{b}_{ 26 } = 1$, $\mathrm{b}_{ 28 } = 1$, $\mathrm{b}_{ 30 } = 1$
$\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{4})$
$\mathrm{F}_4$
$\pi_0\mathrm{Aut}(\mathrm{F}_{4}/\mathrm{P}_{4})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{F}_{4}/\mathrm{P}_{4})$
52
Projective geometry
minimal embedding

$\mathrm{F}_{4}/\mathrm{P}_{4}\hookrightarrow\mathbb{P}^{ 25 }$

degree
78
Hilbert series
1, 26, 324, 2652, 16302, 81081, 342056, 1264120, 4188834, 12664184, 35405968, 92512368, 227854536, 532703874, 1189056024, 2546364040, 5253305915, 10477865970, 20265831300, 38111646300, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
6
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 $\mathbf D^{\mathrm b}(\mathrm{F}_{4}/\mathrm{P}_{4})$:

Quantum cohomology
small
not generically semisimple [Galkin–Leung–Li–Xiong, Theorem 1.2]

Graded dimensions modulo the index $11$: $\widetilde b_{ 4 }=3>2=\widetilde b_{ 8 }$, where $8\equiv 2\cdot 4\pmod{ 11 }$.

big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Earlier proof for coadjoint varieties: [Perrin–Smirnov] .

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{F}_{4}/\mathrm{P}_{4})$:

Homological projective duality

Full minimal nonrectangular Lefschetz collection. [Belmans–Kuznetsov–Smirnov, Theorem 1.1]

Lefschetz block sizes: 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Block sizes $(3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2)$; 24 objects.

Restriction of the Cayley-plane collection to a general hyperplane section.

No HP-dual model recorded.

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{F}_{4}/\mathrm{P}_{4} })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [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. Pieter Belmans; Alexander Kuznetsov; Maxim Smirnov. Derived categories of the Cayley plane and the coadjoint Grassmannian of type F. Transform. Groups 28 (2023), 9–34. arXiv:2005.01989.
  3. 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.
  4. Sergey Galkin; Naichung Conan Leung; Changzheng Li; Rui Xiong; appendix by Pieter Belmans and Markus Reineke. A-D-E diagrams, Hodge–Tate hyperplane sections and semisimple quantum cohomology. Preprint (2025). arXiv:2509.01101.
  5. Nicolas Perrin; Maxim Smirnov. On the big quantum cohomology of coadjoint varieties. Selecta Math. 31 (2025), article 30. arXiv:2112.12436.
  6. 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.