\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 E8/P1

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 } &= 3 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 16 } &= 5 \\ \mathrm{b}_{ 18 } &= 5 \\ \mathrm{b}_{ 20 } &= 7 \\ \mathrm{b}_{ 22 } &= 8 \\ \mathrm{b}_{ 24 } &= 10 \\ \mathrm{b}_{ 26 } &= 11 \\ \mathrm{b}_{ 28 } &= 13 \\ \mathrm{b}_{ 30 } &= 14 \\ \mathrm{b}_{ 32 } &= 17 \\ \mathrm{b}_{ 34 } &= 19 \\ \mathrm{b}_{ 36 } &= 21 \\ \mathrm{b}_{ 38 } &= 23 \\ \mathrm{b}_{ 40 } &= 26 \\ \mathrm{b}_{ 42 } &= 28 \\ \mathrm{b}_{ 44 } &= 31 \\ \mathrm{b}_{ 46 } &= 34 \\ \mathrm{b}_{ 48 } &= 36 \\ \mathrm{b}_{ 50 } &= 38 \\ \mathrm{b}_{ 52 } &= 41 \\ \mathrm{b}_{ 54 } &= 44 \\ \mathrm{b}_{ 56 } &= 46 \\ \mathrm{b}_{ 58 } &= 49 \\ \mathrm{b}_{ 60 } &= 50 \\ \mathrm{b}_{ 62 } &= 52 \\ \mathrm{b}_{ 64 } &= 54 \\ \mathrm{b}_{ 66 } &= 57 \\ \mathrm{b}_{ 68 } &= 57 \\ \mathrm{b}_{ 70 } &= 59 \\ \mathrm{b}_{ 72 } &= 59 \\ \mathrm{b}_{ 74 } &= 60 \\ \mathrm{b}_{ 76 } &= 60 \\ \mathrm{b}_{ 78 } &= 62 \\ \mathrm{b}_{ 80 } &= 60 \\ \mathrm{b}_{ 82 } &= 60 \\ \mathrm{b}_{ 84 } &= 59 \\ \mathrm{b}_{ 86 } &= 59 \\ \mathrm{b}_{ 88 } &= 57 \\ \mathrm{b}_{ 90 } &= 57 \\ \mathrm{b}_{ 92 } &= 54 \\ \mathrm{b}_{ 94 } &= 52 \\ \mathrm{b}_{ 96 } &= 50 \\ \mathrm{b}_{ 98 } &= 49 \\ \mathrm{b}_{ 100 } &= 46 \\ \mathrm{b}_{ 102 } &= 44 \\ \mathrm{b}_{ 104 } &= 41 \\ \mathrm{b}_{ 106 } &= 38 \\ \mathrm{b}_{ 108 } &= 36 \\ \mathrm{b}_{ 110 } &= 34 \\ \mathrm{b}_{ 112 } &= 31 \\ \mathrm{b}_{ 114 } &= 28 \\ \mathrm{b}_{ 116 } &= 26 \\ \mathrm{b}_{ 118 } &= 23 \\ \mathrm{b}_{ 120 } &= 21 \\ \mathrm{b}_{ 122 } &= 19 \\ \mathrm{b}_{ 124 } &= 17 \\ \mathrm{b}_{ 126 } &= 14 \\ \mathrm{b}_{ 128 } &= 13 \\ \mathrm{b}_{ 130 } &= 11 \\ \mathrm{b}_{ 132 } &= 10 \\ \mathrm{b}_{ 134 } &= 8 \\ \mathrm{b}_{ 136 } &= 7 \\ \mathrm{b}_{ 138 } &= 5 \\ \mathrm{b}_{ 140 } &= 5 \\ \mathrm{b}_{ 142 } &= 4 \\ \mathrm{b}_{ 144 } &= 3 \\ \mathrm{b}_{ 146 } &= 2 \\ \mathrm{b}_{ 148 } &= 2 \\ \mathrm{b}_{ 150 } &= 1 \\ \mathrm{b}_{ 152 } &= 1 \\ \mathrm{b}_{ 154 } &= 1 \\ \mathrm{b}_{ 156 } &= 1 \end{align*}
Basic information
dimension
78
index
23
Euler characteristic
2160
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 1$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 7$, $\mathrm{b}_{ 22 } = 8$, $\mathrm{b}_{ 24 } = 10$, $\mathrm{b}_{ 26 } = 11$, $\mathrm{b}_{ 28 } = 13$, $\mathrm{b}_{ 30 } = 14$, $\mathrm{b}_{ 32 } = 17$, $\mathrm{b}_{ 34 } = 19$, $\mathrm{b}_{ 36 } = 21$, $\mathrm{b}_{ 38 } = 23$, $\mathrm{b}_{ 40 } = 26$, $\mathrm{b}_{ 42 } = 28$, $\mathrm{b}_{ 44 } = 31$, $\mathrm{b}_{ 46 } = 34$, $\mathrm{b}_{ 48 } = 36$, $\mathrm{b}_{ 50 } = 38$, $\mathrm{b}_{ 52 } = 41$, $\mathrm{b}_{ 54 } = 44$, $\mathrm{b}_{ 56 } = 46$, $\mathrm{b}_{ 58 } = 49$, $\mathrm{b}_{ 60 } = 50$, $\mathrm{b}_{ 62 } = 52$, $\mathrm{b}_{ 64 } = 54$, $\mathrm{b}_{ 66 } = 57$, $\mathrm{b}_{ 68 } = 57$, $\mathrm{b}_{ 70 } = 59$, $\mathrm{b}_{ 72 } = 59$, $\mathrm{b}_{ 74 } = 60$, $\mathrm{b}_{ 76 } = 60$, $\mathrm{b}_{ 78 } = 62$, $\mathrm{b}_{ 80 } = 60$, $\mathrm{b}_{ 82 } = 60$, $\mathrm{b}_{ 84 } = 59$, $\mathrm{b}_{ 86 } = 59$, $\mathrm{b}_{ 88 } = 57$, $\mathrm{b}_{ 90 } = 57$, $\mathrm{b}_{ 92 } = 54$, $\mathrm{b}_{ 94 } = 52$, $\mathrm{b}_{ 96 } = 50$, $\mathrm{b}_{ 98 } = 49$, $\mathrm{b}_{ 100 } = 46$, $\mathrm{b}_{ 102 } = 44$, $\mathrm{b}_{ 104 } = 41$, $\mathrm{b}_{ 106 } = 38$, $\mathrm{b}_{ 108 } = 36$, $\mathrm{b}_{ 110 } = 34$, $\mathrm{b}_{ 112 } = 31$, $\mathrm{b}_{ 114 } = 28$, $\mathrm{b}_{ 116 } = 26$, $\mathrm{b}_{ 118 } = 23$, $\mathrm{b}_{ 120 } = 21$, $\mathrm{b}_{ 122 } = 19$, $\mathrm{b}_{ 124 } = 17$, $\mathrm{b}_{ 126 } = 14$, $\mathrm{b}_{ 128 } = 13$, $\mathrm{b}_{ 130 } = 11$, $\mathrm{b}_{ 132 } = 10$, $\mathrm{b}_{ 134 } = 8$, $\mathrm{b}_{ 136 } = 7$, $\mathrm{b}_{ 138 } = 5$, $\mathrm{b}_{ 140 } = 5$, $\mathrm{b}_{ 142 } = 4$, $\mathrm{b}_{ 144 } = 3$, $\mathrm{b}_{ 146 } = 2$, $\mathrm{b}_{ 148 } = 2$, $\mathrm{b}_{ 150 } = 1$, $\mathrm{b}_{ 152 } = 1$, $\mathrm{b}_{ 154 } = 1$, $\mathrm{b}_{ 156 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{1})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{1})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{1})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 3874 }$

degree
566737444875521606631975195475968000
Hilbert series
1, 3875, 4881384, 2903770000, 976217435000, 207272853913752, 30016620324194000, 3138280535780660120, 247491761539737346875, 15243441784308510515625, 754291366788666511287000, 30696344926439269703062500, 1047731612488529774828006250, 30497909649841602914892187500, 767999051805484838339634375000, 16939558438056482585955866100000, 330817811768507722440902447250000, 5774915620620045304684694312022000, 90869541917706942429714014772000000, 1298517132972979360163725684963200000, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
2062897335840485479678459920
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 $\mathbf D^{\mathrm b}(\mathrm{E}_{8}/\mathrm{P}_{1})$:

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

Graded dimensions modulo the index $23$: $\widetilde b_{ 1 }=94>93=\widetilde b_{ 2 }$, where $2\equiv 2\cdot 1\pmod{ 23 }$.

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

Convergent near the large-radius limit.

Homological projective duality

No Lefschetz decomposition recorded.

No HP-dual model recorded.

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \mathrm{E}_{8}/\mathrm{P}_{1} })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [Araujo et al., Theorems 1.3 and 1.4]

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. 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.