\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/P2

Betti numbers
\begin{align*} \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 } &= 6 \\ \mathrm{b}_{ 14 } &= 8 \\ \mathrm{b}_{ 16 } &= 10 \\ \mathrm{b}_{ 18 } &= 13 \\ \mathrm{b}_{ 20 } &= 17 \\ \mathrm{b}_{ 22 } &= 21 \\ \mathrm{b}_{ 24 } &= 26 \\ \mathrm{b}_{ 26 } &= 32 \\ \mathrm{b}_{ 28 } &= 38 \\ \mathrm{b}_{ 30 } &= 46 \\ \mathrm{b}_{ 32 } &= 55 \\ \mathrm{b}_{ 34 } &= 64 \\ \mathrm{b}_{ 36 } &= 74 \\ \mathrm{b}_{ 38 } &= 86 \\ \mathrm{b}_{ 40 } &= 98 \\ \mathrm{b}_{ 42 } &= 112 \\ \mathrm{b}_{ 44 } &= 127 \\ \mathrm{b}_{ 46 } &= 142 \\ \mathrm{b}_{ 48 } &= 157 \\ \mathrm{b}_{ 50 } &= 175 \\ \mathrm{b}_{ 52 } &= 193 \\ \mathrm{b}_{ 54 } &= 211 \\ \mathrm{b}_{ 56 } &= 230 \\ \mathrm{b}_{ 58 } &= 249 \\ \mathrm{b}_{ 60 } &= 267 \\ \mathrm{b}_{ 62 } &= 287 \\ \mathrm{b}_{ 64 } &= 307 \\ \mathrm{b}_{ 66 } &= 325 \\ \mathrm{b}_{ 68 } &= 343 \\ \mathrm{b}_{ 70 } &= 361 \\ \mathrm{b}_{ 72 } &= 377 \\ \mathrm{b}_{ 74 } &= 393 \\ \mathrm{b}_{ 76 } &= 409 \\ \mathrm{b}_{ 78 } &= 421 \\ \mathrm{b}_{ 80 } &= 432 \\ \mathrm{b}_{ 82 } &= 443 \\ \mathrm{b}_{ 84 } &= 452 \\ \mathrm{b}_{ 86 } &= 458 \\ \mathrm{b}_{ 88 } &= 464 \\ \mathrm{b}_{ 90 } &= 466 \\ \mathrm{b}_{ 92 } &= 466 \\ \mathrm{b}_{ 94 } &= 466 \\ \mathrm{b}_{ 96 } &= 464 \\ \mathrm{b}_{ 98 } &= 458 \\ \mathrm{b}_{ 100 } &= 452 \\ \mathrm{b}_{ 102 } &= 443 \\ \mathrm{b}_{ 104 } &= 432 \\ \mathrm{b}_{ 106 } &= 421 \\ \mathrm{b}_{ 108 } &= 409 \\ \mathrm{b}_{ 110 } &= 393 \\ \mathrm{b}_{ 112 } &= 377 \\ \mathrm{b}_{ 114 } &= 361 \\ \mathrm{b}_{ 116 } &= 343 \\ \mathrm{b}_{ 118 } &= 325 \\ \mathrm{b}_{ 120 } &= 307 \\ \mathrm{b}_{ 122 } &= 287 \\ \mathrm{b}_{ 124 } &= 267 \\ \mathrm{b}_{ 126 } &= 249 \\ \mathrm{b}_{ 128 } &= 230 \\ \mathrm{b}_{ 130 } &= 211 \\ \mathrm{b}_{ 132 } &= 193 \\ \mathrm{b}_{ 134 } &= 175 \\ \mathrm{b}_{ 136 } &= 157 \\ \mathrm{b}_{ 138 } &= 142 \\ \mathrm{b}_{ 140 } &= 127 \\ \mathrm{b}_{ 142 } &= 112 \\ \mathrm{b}_{ 144 } &= 98 \\ \mathrm{b}_{ 146 } &= 86 \\ \mathrm{b}_{ 148 } &= 74 \\ \mathrm{b}_{ 150 } &= 64 \\ \mathrm{b}_{ 152 } &= 55 \\ \mathrm{b}_{ 154 } &= 46 \\ \mathrm{b}_{ 156 } &= 38 \\ \mathrm{b}_{ 158 } &= 32 \\ \mathrm{b}_{ 160 } &= 26 \\ \mathrm{b}_{ 162 } &= 21 \\ \mathrm{b}_{ 164 } &= 17 \\ \mathrm{b}_{ 166 } &= 13 \\ \mathrm{b}_{ 168 } &= 10 \\ \mathrm{b}_{ 170 } &= 8 \\ \mathrm{b}_{ 172 } &= 6 \\ \mathrm{b}_{ 174 } &= 4 \\ \mathrm{b}_{ 176 } &= 3 \\ \mathrm{b}_{ 178 } &= 2 \\ \mathrm{b}_{ 180 } &= 1 \\ \mathrm{b}_{ 182 } &= 1 \\ \mathrm{b}_{ 184 } &= 1 \end{align*}
Basic information
dimension
92
index
17
Euler characteristic
17280
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 4$, $\mathrm{b}_{ 12 } = 6$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 10$, $\mathrm{b}_{ 18 } = 13$, $\mathrm{b}_{ 20 } = 17$, $\mathrm{b}_{ 22 } = 21$, $\mathrm{b}_{ 24 } = 26$, $\mathrm{b}_{ 26 } = 32$, $\mathrm{b}_{ 28 } = 38$, $\mathrm{b}_{ 30 } = 46$, $\mathrm{b}_{ 32 } = 55$, $\mathrm{b}_{ 34 } = 64$, $\mathrm{b}_{ 36 } = 74$, $\mathrm{b}_{ 38 } = 86$, $\mathrm{b}_{ 40 } = 98$, $\mathrm{b}_{ 42 } = 112$, $\mathrm{b}_{ 44 } = 127$, $\mathrm{b}_{ 46 } = 142$, $\mathrm{b}_{ 48 } = 157$, $\mathrm{b}_{ 50 } = 175$, $\mathrm{b}_{ 52 } = 193$, $\mathrm{b}_{ 54 } = 211$, $\mathrm{b}_{ 56 } = 230$, $\mathrm{b}_{ 58 } = 249$, $\mathrm{b}_{ 60 } = 267$, $\mathrm{b}_{ 62 } = 287$, $\mathrm{b}_{ 64 } = 307$, $\mathrm{b}_{ 66 } = 325$, $\mathrm{b}_{ 68 } = 343$, $\mathrm{b}_{ 70 } = 361$, $\mathrm{b}_{ 72 } = 377$, $\mathrm{b}_{ 74 } = 393$, $\mathrm{b}_{ 76 } = 409$, $\mathrm{b}_{ 78 } = 421$, $\mathrm{b}_{ 80 } = 432$, $\mathrm{b}_{ 82 } = 443$, $\mathrm{b}_{ 84 } = 452$, $\mathrm{b}_{ 86 } = 458$, $\mathrm{b}_{ 88 } = 464$, $\mathrm{b}_{ 90 } = 466$, $\mathrm{b}_{ 92 } = 466$, $\mathrm{b}_{ 94 } = 466$, $\mathrm{b}_{ 96 } = 464$, $\mathrm{b}_{ 98 } = 458$, $\mathrm{b}_{ 100 } = 452$, $\mathrm{b}_{ 102 } = 443$, $\mathrm{b}_{ 104 } = 432$, $\mathrm{b}_{ 106 } = 421$, $\mathrm{b}_{ 108 } = 409$, $\mathrm{b}_{ 110 } = 393$, $\mathrm{b}_{ 112 } = 377$, $\mathrm{b}_{ 114 } = 361$, $\mathrm{b}_{ 116 } = 343$, $\mathrm{b}_{ 118 } = 325$, $\mathrm{b}_{ 120 } = 307$, $\mathrm{b}_{ 122 } = 287$, $\mathrm{b}_{ 124 } = 267$, $\mathrm{b}_{ 126 } = 249$, $\mathrm{b}_{ 128 } = 230$, $\mathrm{b}_{ 130 } = 211$, $\mathrm{b}_{ 132 } = 193$, $\mathrm{b}_{ 134 } = 175$, $\mathrm{b}_{ 136 } = 157$, $\mathrm{b}_{ 138 } = 142$, $\mathrm{b}_{ 140 } = 127$, $\mathrm{b}_{ 142 } = 112$, $\mathrm{b}_{ 144 } = 98$, $\mathrm{b}_{ 146 } = 86$, $\mathrm{b}_{ 148 } = 74$, $\mathrm{b}_{ 150 } = 64$, $\mathrm{b}_{ 152 } = 55$, $\mathrm{b}_{ 154 } = 46$, $\mathrm{b}_{ 156 } = 38$, $\mathrm{b}_{ 158 } = 32$, $\mathrm{b}_{ 160 } = 26$, $\mathrm{b}_{ 162 } = 21$, $\mathrm{b}_{ 164 } = 17$, $\mathrm{b}_{ 166 } = 13$, $\mathrm{b}_{ 168 } = 10$, $\mathrm{b}_{ 170 } = 8$, $\mathrm{b}_{ 172 } = 6$, $\mathrm{b}_{ 174 } = 4$, $\mathrm{b}_{ 176 } = 3$, $\mathrm{b}_{ 178 } = 2$, $\mathrm{b}_{ 180 } = 1$, $\mathrm{b}_{ 182 } = 1$, $\mathrm{b}_{ 184 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{2})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{2})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{2})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 147249 }$

degree
221742396458192785546942875610662293134814379227347943424000
Hilbert series
1, 147250, 4076399250, 35361935272950, 125982160881890625, 219535295360061727800, 211712396994333259128000, 123868068014021389904832000, 47200473064255798138436218368, 12390986693062787651882950840000, 2345118041783809242861398688412888, 332154157389659136763618228106828625, 36320209923251219664609477939100007500, 3147687810989929917787834478573310805500, 221097506373270548223283132912819251562500, 12831460733037551669320439047215561510234375, 625611363932726266078933184258231580175781250, 26000628460280646817980180919271683490332031250, 932950905359274517791652426815027754329677812500, 29229869285264902491764188998651481144840509375000, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
720390638145654091106819793484077346214538742611235200
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 $\mathbf D^{\mathrm b}(\mathrm{E}_{8}/\mathrm{P}_{2})$:

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

Graded dimensions modulo the index $17$: $\widetilde b_{ 3 }=1017>1016=\widetilde b_{ 0 }$, where $0\equiv 0\cdot 3\pmod{ 17 }$.

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

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.