\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/P4

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 3 \\ \mathrm{b}_{ 6 } &= 5 \\ \mathrm{b}_{ 8 } &= 9 \\ \mathrm{b}_{ 10 } &= 14 \\ \mathrm{b}_{ 12 } &= 23 \\ \mathrm{b}_{ 14 } &= 33 \\ \mathrm{b}_{ 16 } &= 49 \\ \mathrm{b}_{ 18 } &= 69 \\ \mathrm{b}_{ 20 } &= 96 \\ \mathrm{b}_{ 22 } &= 129 \\ \mathrm{b}_{ 24 } &= 173 \\ \mathrm{b}_{ 26 } &= 225 \\ \mathrm{b}_{ 28 } &= 290 \\ \mathrm{b}_{ 30 } &= 369 \\ \mathrm{b}_{ 32 } &= 462 \\ \mathrm{b}_{ 34 } &= 572 \\ \mathrm{b}_{ 36 } &= 701 \\ \mathrm{b}_{ 38 } &= 850 \\ \mathrm{b}_{ 40 } &= 1019 \\ \mathrm{b}_{ 42 } &= 1214 \\ \mathrm{b}_{ 44 } &= 1430 \\ \mathrm{b}_{ 46 } &= 1673 \\ \mathrm{b}_{ 48 } &= 1940 \\ \mathrm{b}_{ 50 } &= 2237 \\ \mathrm{b}_{ 52 } &= 2555 \\ \mathrm{b}_{ 54 } &= 2905 \\ \mathrm{b}_{ 56 } &= 3276 \\ \mathrm{b}_{ 58 } &= 3675 \\ \mathrm{b}_{ 60 } &= 4094 \\ \mathrm{b}_{ 62 } &= 4540 \\ \mathrm{b}_{ 64 } &= 4999 \\ \mathrm{b}_{ 66 } &= 5480 \\ \mathrm{b}_{ 68 } &= 5972 \\ \mathrm{b}_{ 70 } &= 6477 \\ \mathrm{b}_{ 72 } &= 6986 \\ \mathrm{b}_{ 74 } &= 7503 \\ \mathrm{b}_{ 76 } &= 8015 \\ \mathrm{b}_{ 78 } &= 8523 \\ \mathrm{b}_{ 80 } &= 9023 \\ \mathrm{b}_{ 82 } &= 9508 \\ \mathrm{b}_{ 84 } &= 9974 \\ \mathrm{b}_{ 86 } &= 10418 \\ \mathrm{b}_{ 88 } &= 10836 \\ \mathrm{b}_{ 90 } &= 11218 \\ \mathrm{b}_{ 92 } &= 11571 \\ \mathrm{b}_{ 94 } &= 11881 \\ \mathrm{b}_{ 96 } &= 12151 \\ \mathrm{b}_{ 98 } &= 12373 \\ \mathrm{b}_{ 100 } &= 12554 \\ \mathrm{b}_{ 102 } &= 12677 \\ \mathrm{b}_{ 104 } &= 12758 \\ \mathrm{b}_{ 106 } &= 12782 \\ \mathrm{b}_{ 108 } &= 12758 \\ \mathrm{b}_{ 110 } &= 12677 \\ \mathrm{b}_{ 112 } &= 12554 \\ \mathrm{b}_{ 114 } &= 12373 \\ \mathrm{b}_{ 116 } &= 12151 \\ \mathrm{b}_{ 118 } &= 11881 \\ \mathrm{b}_{ 120 } &= 11571 \\ \mathrm{b}_{ 122 } &= 11218 \\ \mathrm{b}_{ 124 } &= 10836 \\ \mathrm{b}_{ 126 } &= 10418 \\ \mathrm{b}_{ 128 } &= 9974 \\ \mathrm{b}_{ 130 } &= 9508 \\ \mathrm{b}_{ 132 } &= 9023 \\ \mathrm{b}_{ 134 } &= 8523 \\ \mathrm{b}_{ 136 } &= 8015 \\ \mathrm{b}_{ 138 } &= 7503 \\ \mathrm{b}_{ 140 } &= 6986 \\ \mathrm{b}_{ 142 } &= 6477 \\ \mathrm{b}_{ 144 } &= 5972 \\ \mathrm{b}_{ 146 } &= 5480 \\ \mathrm{b}_{ 148 } &= 4999 \\ \mathrm{b}_{ 150 } &= 4540 \\ \mathrm{b}_{ 152 } &= 4094 \\ \mathrm{b}_{ 154 } &= 3675 \\ \mathrm{b}_{ 156 } &= 3276 \\ \mathrm{b}_{ 158 } &= 2905 \\ \mathrm{b}_{ 160 } &= 2555 \\ \mathrm{b}_{ 162 } &= 2237 \\ \mathrm{b}_{ 164 } &= 1940 \\ \mathrm{b}_{ 166 } &= 1673 \\ \mathrm{b}_{ 168 } &= 1430 \\ \mathrm{b}_{ 170 } &= 1214 \\ \mathrm{b}_{ 172 } &= 1019 \\ \mathrm{b}_{ 174 } &= 850 \\ \mathrm{b}_{ 176 } &= 701 \\ \mathrm{b}_{ 178 } &= 572 \\ \mathrm{b}_{ 180 } &= 462 \\ \mathrm{b}_{ 182 } &= 369 \\ \mathrm{b}_{ 184 } &= 290 \\ \mathrm{b}_{ 186 } &= 225 \\ \mathrm{b}_{ 188 } &= 173 \\ \mathrm{b}_{ 190 } &= 129 \\ \mathrm{b}_{ 192 } &= 96 \\ \mathrm{b}_{ 194 } &= 69 \\ \mathrm{b}_{ 196 } &= 49 \\ \mathrm{b}_{ 198 } &= 33 \\ \mathrm{b}_{ 200 } &= 23 \\ \mathrm{b}_{ 202 } &= 14 \\ \mathrm{b}_{ 204 } &= 9 \\ \mathrm{b}_{ 206 } &= 5 \\ \mathrm{b}_{ 208 } &= 3 \\ \mathrm{b}_{ 210 } &= 1 \\ \mathrm{b}_{ 212 } &= 1 \end{align*}
Basic information
dimension
106
index
9
Euler characteristic
483840
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 3$, $\mathrm{b}_{ 6 } = 5$, $\mathrm{b}_{ 8 } = 9$, $\mathrm{b}_{ 10 } = 14$, $\mathrm{b}_{ 12 } = 23$, $\mathrm{b}_{ 14 } = 33$, $\mathrm{b}_{ 16 } = 49$, $\mathrm{b}_{ 18 } = 69$, $\mathrm{b}_{ 20 } = 96$, $\mathrm{b}_{ 22 } = 129$, $\mathrm{b}_{ 24 } = 173$, $\mathrm{b}_{ 26 } = 225$, $\mathrm{b}_{ 28 } = 290$, $\mathrm{b}_{ 30 } = 369$, $\mathrm{b}_{ 32 } = 462$, $\mathrm{b}_{ 34 } = 572$, $\mathrm{b}_{ 36 } = 701$, $\mathrm{b}_{ 38 } = 850$, $\mathrm{b}_{ 40 } = 1019$, $\mathrm{b}_{ 42 } = 1214$, $\mathrm{b}_{ 44 } = 1430$, $\mathrm{b}_{ 46 } = 1673$, $\mathrm{b}_{ 48 } = 1940$, $\mathrm{b}_{ 50 } = 2237$, $\mathrm{b}_{ 52 } = 2555$, $\mathrm{b}_{ 54 } = 2905$, $\mathrm{b}_{ 56 } = 3276$, $\mathrm{b}_{ 58 } = 3675$, $\mathrm{b}_{ 60 } = 4094$, $\mathrm{b}_{ 62 } = 4540$, $\mathrm{b}_{ 64 } = 4999$, $\mathrm{b}_{ 66 } = 5480$, $\mathrm{b}_{ 68 } = 5972$, $\mathrm{b}_{ 70 } = 6477$, $\mathrm{b}_{ 72 } = 6986$, $\mathrm{b}_{ 74 } = 7503$, $\mathrm{b}_{ 76 } = 8015$, $\mathrm{b}_{ 78 } = 8523$, $\mathrm{b}_{ 80 } = 9023$, $\mathrm{b}_{ 82 } = 9508$, $\mathrm{b}_{ 84 } = 9974$, $\mathrm{b}_{ 86 } = 10418$, $\mathrm{b}_{ 88 } = 10836$, $\mathrm{b}_{ 90 } = 11218$, $\mathrm{b}_{ 92 } = 11571$, $\mathrm{b}_{ 94 } = 11881$, $\mathrm{b}_{ 96 } = 12151$, $\mathrm{b}_{ 98 } = 12373$, $\mathrm{b}_{ 100 } = 12554$, $\mathrm{b}_{ 102 } = 12677$, $\mathrm{b}_{ 104 } = 12758$, $\mathrm{b}_{ 106 } = 12782$, $\mathrm{b}_{ 108 } = 12758$, $\mathrm{b}_{ 110 } = 12677$, $\mathrm{b}_{ 112 } = 12554$, $\mathrm{b}_{ 114 } = 12373$, $\mathrm{b}_{ 116 } = 12151$, $\mathrm{b}_{ 118 } = 11881$, $\mathrm{b}_{ 120 } = 11571$, $\mathrm{b}_{ 122 } = 11218$, $\mathrm{b}_{ 124 } = 10836$, $\mathrm{b}_{ 126 } = 10418$, $\mathrm{b}_{ 128 } = 9974$, $\mathrm{b}_{ 130 } = 9508$, $\mathrm{b}_{ 132 } = 9023$, $\mathrm{b}_{ 134 } = 8523$, $\mathrm{b}_{ 136 } = 8015$, $\mathrm{b}_{ 138 } = 7503$, $\mathrm{b}_{ 140 } = 6986$, $\mathrm{b}_{ 142 } = 6477$, $\mathrm{b}_{ 144 } = 5972$, $\mathrm{b}_{ 146 } = 5480$, $\mathrm{b}_{ 148 } = 4999$, $\mathrm{b}_{ 150 } = 4540$, $\mathrm{b}_{ 152 } = 4094$, $\mathrm{b}_{ 154 } = 3675$, $\mathrm{b}_{ 156 } = 3276$, $\mathrm{b}_{ 158 } = 2905$, $\mathrm{b}_{ 160 } = 2555$, $\mathrm{b}_{ 162 } = 2237$, $\mathrm{b}_{ 164 } = 1940$, $\mathrm{b}_{ 166 } = 1673$, $\mathrm{b}_{ 168 } = 1430$, $\mathrm{b}_{ 170 } = 1214$, $\mathrm{b}_{ 172 } = 1019$, $\mathrm{b}_{ 174 } = 850$, $\mathrm{b}_{ 176 } = 701$, $\mathrm{b}_{ 178 } = 572$, $\mathrm{b}_{ 180 } = 462$, $\mathrm{b}_{ 182 } = 369$, $\mathrm{b}_{ 184 } = 290$, $\mathrm{b}_{ 186 } = 225$, $\mathrm{b}_{ 188 } = 173$, $\mathrm{b}_{ 190 } = 129$, $\mathrm{b}_{ 192 } = 96$, $\mathrm{b}_{ 194 } = 69$, $\mathrm{b}_{ 196 } = 49$, $\mathrm{b}_{ 198 } = 33$, $\mathrm{b}_{ 200 } = 23$, $\mathrm{b}_{ 202 } = 14$, $\mathrm{b}_{ 204 } = 9$, $\mathrm{b}_{ 206 } = 5$, $\mathrm{b}_{ 208 } = 3$, $\mathrm{b}_{ 210 } = 1$, $\mathrm{b}_{ 212 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{4})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{4})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{4})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{4}\hookrightarrow\mathbb{P}^{ 6899079263 }$

degree
2602864548591401547153855086983839499833287935068781595086354415765978264937928543198815191040000000000
Hilbert series
1, 6899079264, 684252595649750400, 4020258593733507722562560, 3044046738855704109445572480000, 483365215657555731024878991971987136, 22358313500264160459723570009871204977664, 380441968987367390348876814080876331373520500, 2828672569442851162646738736520484513149992187500, 10474617057593062950836429764942122099110650934343000, 21389046901090905363927828451370456474013399656559207936, 26114574684023345236921648731612366481268083014749627875328, 20351915250112209875432258271879679311427596789203431587840000, 10681578808397170382541409070096040180295815167553004502935319040, 3947294888183787977279714898856304404599137090659020372088123168200, 1066160447128916625983499671987813359535079850519920025015252205187858, 217250226701864118226450206410586974682113070836753913641055073363143120, 34314496758213991074499645872359903313044555376368599844938311069656973312, 4300476414986241602583208498304859439203919848748600903217393093575170486400, 436397108255712106051994455438541565358650335440809857352813982855800752000000, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
35611950426680262686378600395596539274124653801922433913017963258803265832534463674435968918952829440
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 $\mathbf D^{\mathrm b}(\mathrm{E}_{8}/\mathrm{P}_{4})$:

Quantum cohomology
small
not generically semisimple [Chaput–Perrin, table on p. 326]
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

Fano but not 2-Fano. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Chaput, P. E.; Perrin, N. On the quantum cohomology of adjoint varieties. Proc. Lond. Math. Soc. (3) 103 (2011), 294–330. arXiv:0904.4824. MR2821244.
  2. 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.
  3. 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.
  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.