Grassmannian.info

dimension

42

index

14

Euler characteristic

576

Betti numbers

$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 4$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 14 } = 7$, $\mathrm{b}_{ 16 } = 8$, $\mathrm{b}_{ 18 } = 10$, $\mathrm{b}_{ 20 } = 12$, $\mathrm{b}_{ 22 } = 14$, $\mathrm{b}_{ 24 } = 16$, $\mathrm{b}_{ 26 } = 18$, $\mathrm{b}_{ 28 } = 20$, $\mathrm{b}_{ 30 } = 22$, $\mathrm{b}_{ 32 } = 24$, $\mathrm{b}_{ 34 } = 25$, $\mathrm{b}_{ 36 } = 26$, $\mathrm{b}_{ 38 } = 27$, $\mathrm{b}_{ 40 } = 28$, $\mathrm{b}_{ 42 } = 28$, $\mathrm{b}_{ 44 } = 28$, $\mathrm{b}_{ 46 } = 27$, $\mathrm{b}_{ 48 } = 26$, $\mathrm{b}_{ 50 } = 25$, $\mathrm{b}_{ 52 } = 24$, $\mathrm{b}_{ 54 } = 22$, $\mathrm{b}_{ 56 } = 20$, $\mathrm{b}_{ 58 } = 18$, $\mathrm{b}_{ 60 } = 16$, $\mathrm{b}_{ 62 } = 14$, $\mathrm{b}_{ 64 } = 12$, $\mathrm{b}_{ 66 } = 10$, $\mathrm{b}_{ 68 } = 8$, $\mathrm{b}_{ 70 } = 7$, $\mathrm{b}_{ 72 } = 5$, $\mathrm{b}_{ 74 } = 4$, $\mathrm{b}_{ 76 } = 3$, $\mathrm{b}_{ 78 } = 2$, $\mathrm{b}_{ 80 } = 1$, $\mathrm{b}_{ 82 } = 1$, $\mathrm{b}_{ 84 } = 1$

$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{2})$

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

$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{2})$

$1$

$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{2})$

133

minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 911 }$

degree

230629602357872640

Hilbert series

1, 912, 253935, 32995248, 2457458575, 118861039440, 4061118864660, 104114789500800, 2095350182853408, 34279106505180160, 468620731551643872, 5474059247498898816, 55645606128266708700, 499788275805501361360, 4017247956270188974125, 29213221999317397602000, 193993785997407232711875, 1185918539273268050250000, 6720932040786435334078125, 35529086969201421823500000, ...

The projective dual is a hypersurface.

codimension

1

degree

5756196004320

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 $\mathbf D^{\mathrm b}(\mathrm{E}_{7}/\mathrm{P}_{2})$:

• Samokhin–van der Kallen (2024)

small

generically semisimple [Holmes–Muratore, Theorems 1.2 and 6.2; Remark 6.3]

Semisimple at $q=1$; $c_1\star$ has the eigenvalue $0$ with algebraic multiplicity 2.

big

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

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{7}/\mathrm{P}_{2})$:

No Lefschetz decomposition recorded.

No HP-dual model recorded.

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{E}_{7}/\mathrm{P}_{2} })\cdot Z>0$ 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. Daniel Holmes; Giosuè Muratore. Effective Gromov–Witten computations for GKM spaces and a semisimple application. Preprint (2026). arXiv:2610.37273.
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.