# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type E8/P4

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, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{4})$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is not known yet to be generically semisimple.

Homological projective duality