# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Generalised Grassmannian of type E8/P3

Basic information
dimension
98
index
13
Euler characteristic
69120
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 7$, $\mathrm{b}_{ 12 } = 11$, $\mathrm{b}_{ 14 } = 15$, $\mathrm{b}_{ 16 } = 20$, $\mathrm{b}_{ 18 } = 27$, $\mathrm{b}_{ 20 } = 36$, $\mathrm{b}_{ 22 } = 46$, $\mathrm{b}_{ 24 } = 59$, $\mathrm{b}_{ 26 } = 74$, $\mathrm{b}_{ 28 } = 91$, $\mathrm{b}_{ 30 } = 112$, $\mathrm{b}_{ 32 } = 136$, $\mathrm{b}_{ 34 } = 163$, $\mathrm{b}_{ 36 } = 193$, $\mathrm{b}_{ 38 } = 228$, $\mathrm{b}_{ 40 } = 265$, $\mathrm{b}_{ 42 } = 308$, $\mathrm{b}_{ 44 } = 354$, $\mathrm{b}_{ 46 } = 404$, $\mathrm{b}_{ 48 } = 456$, $\mathrm{b}_{ 50 } = 515$, $\mathrm{b}_{ 52 } = 575$, $\mathrm{b}_{ 54 } = 640$, $\mathrm{b}_{ 56 } = 707$, $\mathrm{b}_{ 58 } = 777$, $\mathrm{b}_{ 60 } = 847$, $\mathrm{b}_{ 62 } = 922$, $\mathrm{b}_{ 64 } = 997$, $\mathrm{b}_{ 66 } = 1072$, $\mathrm{b}_{ 68 } = 1147$, $\mathrm{b}_{ 70 } = 1222$, $\mathrm{b}_{ 72 } = 1294$, $\mathrm{b}_{ 74 } = 1366$, $\mathrm{b}_{ 76 } = 1436$, $\mathrm{b}_{ 78 } = 1500$, $\mathrm{b}_{ 80 } = 1561$, $\mathrm{b}_{ 82 } = 1618$, $\mathrm{b}_{ 84 } = 1670$, $\mathrm{b}_{ 86 } = 1715$, $\mathrm{b}_{ 88 } = 1757$, $\mathrm{b}_{ 90 } = 1788$, $\mathrm{b}_{ 92 } = 1814$, $\mathrm{b}_{ 94 } = 1833$, $\mathrm{b}_{ 96 } = 1846$, $\mathrm{b}_{ 98 } = 1848$, $\mathrm{b}_{ 100 } = 1846$, $\mathrm{b}_{ 102 } = 1833$, $\mathrm{b}_{ 104 } = 1814$, $\mathrm{b}_{ 106 } = 1788$, $\mathrm{b}_{ 108 } = 1757$, $\mathrm{b}_{ 110 } = 1715$, $\mathrm{b}_{ 112 } = 1670$, $\mathrm{b}_{ 114 } = 1618$, $\mathrm{b}_{ 116 } = 1561$, $\mathrm{b}_{ 118 } = 1500$, $\mathrm{b}_{ 120 } = 1436$, $\mathrm{b}_{ 122 } = 1366$, $\mathrm{b}_{ 124 } = 1294$, $\mathrm{b}_{ 126 } = 1222$, $\mathrm{b}_{ 128 } = 1147$, $\mathrm{b}_{ 130 } = 1072$, $\mathrm{b}_{ 132 } = 997$, $\mathrm{b}_{ 134 } = 922$, $\mathrm{b}_{ 136 } = 847$, $\mathrm{b}_{ 138 } = 777$, $\mathrm{b}_{ 140 } = 707$, $\mathrm{b}_{ 142 } = 640$, $\mathrm{b}_{ 144 } = 575$, $\mathrm{b}_{ 146 } = 515$, $\mathrm{b}_{ 148 } = 456$, $\mathrm{b}_{ 150 } = 404$, $\mathrm{b}_{ 152 } = 354$, $\mathrm{b}_{ 154 } = 308$, $\mathrm{b}_{ 156 } = 265$, $\mathrm{b}_{ 158 } = 228$, $\mathrm{b}_{ 160 } = 193$, $\mathrm{b}_{ 162 } = 163$, $\mathrm{b}_{ 164 } = 136$, $\mathrm{b}_{ 166 } = 112$, $\mathrm{b}_{ 168 } = 91$, $\mathrm{b}_{ 170 } = 74$, $\mathrm{b}_{ 172 } = 59$, $\mathrm{b}_{ 174 } = 46$, $\mathrm{b}_{ 176 } = 36$, $\mathrm{b}_{ 178 } = 27$, $\mathrm{b}_{ 180 } = 20$, $\mathrm{b}_{ 182 } = 15$, $\mathrm{b}_{ 184 } = 11$, $\mathrm{b}_{ 186 } = 7$, $\mathrm{b}_{ 188 } = 5$, $\mathrm{b}_{ 190 } = 3$, $\mathrm{b}_{ 192 } = 2$, $\mathrm{b}_{ 194 } = 1$, $\mathrm{b}_{ 196 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{3})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{3})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{3})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{3}\hookrightarrow\mathbb{P}^{ 6695999 }$

degree
50977565117072727424953142274814106015982560278817610815084161391134769152000
Hilbert series
1, 6696000, 3754721200320, 381685932161088750, 10736931073672203345000, 109749414417460376126568000, 492857856408576376994810625000, 1117446409714022991737184491883000, 1421471992649065400418340532083920000, 1101475531614828226176100972502384640000, 555117888165451513731968763477876112097280, 191881555354969234862289927082276765783824000, 47522345316896591579201674566318193209956827752, 8745758901567026733656095134195705920608246695000, 1233284529098589196497616200440126735909933655840000, 136785542814570908676045353258124057098191018434066400, 12203002848388693229611643211411984515345851688093062500, 892826408680706909922092309733699304691677168692615000000, 54485283872355489956051960674271359127333503517578125000000, 2814722355713562257611652433995186567978593302743865966796875, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{3})$. 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