Grassmannian.info

dimension

57

index

29

Euler characteristic

240

Betti numbers

$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 1$, $\mathrm{b}_{ 10 } = 1$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 3$, $\mathrm{b}_{ 22 } = 3$, $\mathrm{b}_{ 24 } = 4$, $\mathrm{b}_{ 26 } = 4$, $\mathrm{b}_{ 28 } = 4$, $\mathrm{b}_{ 30 } = 4$, $\mathrm{b}_{ 32 } = 5$, $\mathrm{b}_{ 34 } = 5$, $\mathrm{b}_{ 36 } = 6$, $\mathrm{b}_{ 38 } = 6$, $\mathrm{b}_{ 40 } = 6$, $\mathrm{b}_{ 42 } = 6$, $\mathrm{b}_{ 44 } = 7$, $\mathrm{b}_{ 46 } = 7$, $\mathrm{b}_{ 48 } = 7$, $\mathrm{b}_{ 50 } = 7$, $\mathrm{b}_{ 52 } = 7$, $\mathrm{b}_{ 54 } = 7$, $\mathrm{b}_{ 56 } = 8$, $\mathrm{b}_{ 58 } = 8$, $\mathrm{b}_{ 60 } = 7$, $\mathrm{b}_{ 62 } = 7$, $\mathrm{b}_{ 64 } = 7$, $\mathrm{b}_{ 66 } = 7$, $\mathrm{b}_{ 68 } = 7$, $\mathrm{b}_{ 70 } = 7$, $\mathrm{b}_{ 72 } = 6$, $\mathrm{b}_{ 74 } = 6$, $\mathrm{b}_{ 76 } = 6$, $\mathrm{b}_{ 78 } = 6$, $\mathrm{b}_{ 80 } = 5$, $\mathrm{b}_{ 82 } = 5$, $\mathrm{b}_{ 84 } = 4$, $\mathrm{b}_{ 86 } = 4$, $\mathrm{b}_{ 88 } = 4$, $\mathrm{b}_{ 90 } = 4$, $\mathrm{b}_{ 92 } = 3$, $\mathrm{b}_{ 94 } = 3$, $\mathrm{b}_{ 96 } = 2$, $\mathrm{b}_{ 98 } = 2$, $\mathrm{b}_{ 100 } = 2$, $\mathrm{b}_{ 102 } = 2$, $\mathrm{b}_{ 104 } = 1$, $\mathrm{b}_{ 106 } = 1$, $\mathrm{b}_{ 108 } = 1$, $\mathrm{b}_{ 110 } = 1$, $\mathrm{b}_{ 112 } = 1$, $\mathrm{b}_{ 114 } = 1$

$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{8})$

adjoint group of type $\mathrm{E}_{ 8 }$

$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{8})$

$1$

$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{8})$

248

minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{8}\hookrightarrow\mathbb{P}^{ 247 }$

degree

126937516885200

Hilbert series

1, 248, 27000, 1763125, 79143000, 2642777280, 69176971200, 1473701482500, 26284473168750, 401283501480000, 5338265882241600, 62790857238950100, 661062273763905000, 6294003651511200000, 54675736068345120000, 436687003868825311200, 3228153165040477279320, 22217485351372039512000, 143102432756681687640000, 866595309136135835343000, ...

The projective dual is a hypersurface.

codimension

1

degree

240

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 $\mathbf D^{\mathrm b}(\mathrm{E}_{8}/\mathrm{P}_{8})$:

• Samokhin–van der Kallen (2024)

small

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

Graded dimensions modulo the index $29$: $\widetilde b_{ 6 }=9>8=\widetilde b_{ 24 }$, where $24\equiv 4\cdot 6\pmod{ 29 }$.

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 $\mathrm{c}_1(\mathrm{E}_{8}/\mathrm{P}_{8})$:

No Lefschetz decomposition recorded.

No HP-dual model recorded.

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{E}_{8}/\mathrm{P}_{8} })\cdot Z>0$ 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.