Grassmannian.info

dimension

33

index

17

Euler characteristic

126

Betti numbers

$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 14 } = 3$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 18 } = 4$, $\mathrm{b}_{ 20 } = 5$, $\mathrm{b}_{ 22 } = 5$, $\mathrm{b}_{ 24 } = 6$, $\mathrm{b}_{ 26 } = 6$, $\mathrm{b}_{ 28 } = 6$, $\mathrm{b}_{ 30 } = 6$, $\mathrm{b}_{ 32 } = 7$, $\mathrm{b}_{ 34 } = 7$, $\mathrm{b}_{ 36 } = 6$, $\mathrm{b}_{ 38 } = 6$, $\mathrm{b}_{ 40 } = 6$, $\mathrm{b}_{ 42 } = 6$, $\mathrm{b}_{ 44 } = 5$, $\mathrm{b}_{ 46 } = 5$, $\mathrm{b}_{ 48 } = 4$, $\mathrm{b}_{ 50 } = 4$, $\mathrm{b}_{ 52 } = 3$, $\mathrm{b}_{ 54 } = 3$, $\mathrm{b}_{ 56 } = 2$, $\mathrm{b}_{ 58 } = 2$, $\mathrm{b}_{ 60 } = 1$, $\mathrm{b}_{ 62 } = 1$, $\mathrm{b}_{ 64 } = 1$, $\mathrm{b}_{ 66 } = 1$

$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{1})$

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

$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{1})$

$1$

$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{1})$

133

minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 132 }$

degree

141430680

Hilbert series

1, 133, 7371, 238602, 5248750, 85709988, 1101296924, 11604306012, 103402141164, 797856027500, 5431803835220, 33125614508610, 183226228734150, 928793118827175, 4352687787515625, 18999500104801125, 77742635367237750, 299864450702202750, 1095594009861809250, 3807923569529625000, ...

The projective dual is a hypersurface.

codimension

1

degree

126

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 $\mathbf D^{\mathrm b}(\mathrm{E}_{7}/\mathrm{P}_{1})$:

• Samokhin–van der Kallen (2024)

small

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

Graded dimensions modulo the index $17$: $\widetilde b_{ 4 }=8>7=\widetilde b_{ 3 }$, where $3\equiv 5\cdot 4\pmod{ 17 }$.

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 $\mathrm{c}_1(\mathrm{E}_{7}/\mathrm{P}_{1})$:

No Lefschetz decomposition recorded.

No HP-dual model recorded.

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{E}_{7}/\mathrm{P}_{1} })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [Araujo et al., Theorems 1.3 and 1.4]

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. Nicolas Perrin; Maxim Smirnov. On the big quantum cohomology of coadjoint varieties. Selecta Math. 31 (2025), article 30. arXiv:2112.12436.
5. 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.