Grassmannian.info

dimension

47

index

11

Euler characteristic

2016

Betti numbers

$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 7$, $\mathrm{b}_{ 12 } = 10$, $\mathrm{b}_{ 14 } = 13$, $\mathrm{b}_{ 16 } = 17$, $\mathrm{b}_{ 18 } = 22$, $\mathrm{b}_{ 20 } = 27$, $\mathrm{b}_{ 22 } = 33$, $\mathrm{b}_{ 24 } = 39$, $\mathrm{b}_{ 26 } = 46$, $\mathrm{b}_{ 28 } = 52$, $\mathrm{b}_{ 30 } = 60$, $\mathrm{b}_{ 32 } = 66$, $\mathrm{b}_{ 34 } = 73$, $\mathrm{b}_{ 36 } = 78$, $\mathrm{b}_{ 38 } = 84$, $\mathrm{b}_{ 40 } = 88$, $\mathrm{b}_{ 42 } = 92$, $\mathrm{b}_{ 44 } = 94$, $\mathrm{b}_{ 46 } = 95$, $\mathrm{b}_{ 48 } = 95$, $\mathrm{b}_{ 50 } = 94$, $\mathrm{b}_{ 52 } = 92$, $\mathrm{b}_{ 54 } = 88$, $\mathrm{b}_{ 56 } = 84$, $\mathrm{b}_{ 58 } = 78$, $\mathrm{b}_{ 60 } = 73$, $\mathrm{b}_{ 62 } = 66$, $\mathrm{b}_{ 64 } = 60$, $\mathrm{b}_{ 66 } = 52$, $\mathrm{b}_{ 68 } = 46$, $\mathrm{b}_{ 70 } = 39$, $\mathrm{b}_{ 72 } = 33$, $\mathrm{b}_{ 74 } = 27$, $\mathrm{b}_{ 76 } = 22$, $\mathrm{b}_{ 78 } = 17$, $\mathrm{b}_{ 80 } = 13$, $\mathrm{b}_{ 82 } = 10$, $\mathrm{b}_{ 84 } = 7$, $\mathrm{b}_{ 86 } = 5$, $\mathrm{b}_{ 88 } = 3$, $\mathrm{b}_{ 90 } = 2$, $\mathrm{b}_{ 92 } = 1$, $\mathrm{b}_{ 94 } = 1$

$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{3})$

adjoint group of type $\mathrm{E}_{ 7 }$

$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{3})$

$1$

$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{3})$

133

minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{3}\hookrightarrow\mathbb{P}^{ 8644 }$

degree

135848348778713395019120640

Hilbert series

1, 8645, 13728792, 7323895800, 1778618202669, 236623544427270, 19519433726417625, 1090168539699417750, 43993460888906665500, 1348095176669475954720, 32611758857289095218176, 642443206178278977561600, 10569578762996426893513248, 148276471308422875531385580, 1804732963658859170305997394, 19339069897951186534147208905, 184736647641066570617069001500, 1590064141215173345940892818375, 12446412671545323701870264625000, 89319908088757664551347148125000, ...

The projective dual is a hypersurface.

codimension

1

degree

99290401087531048449312

self-dual

no

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

Full exceptional collections in $\mathbf D^{\mathrm b}(\mathrm{E}_{7}/\mathrm{P}_{3})$:

• Samokhin–van der Kallen (2024)

small

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

Graded dimensions modulo the index $11$: $\widetilde b_{ 1 }=183>182=\widetilde b_{ 7 }$, where $7\equiv 7\cdot 1\pmod{ 11 }$.

big

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

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{7}/\mathrm{P}_{3})$:

No Lefschetz decomposition recorded.

No HP-dual model recorded.

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

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. 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.
4. 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.