\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/P5

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 8 } &= 7 \\ \mathrm{b}_{ 10 } &= 10 \\ \mathrm{b}_{ 12 } &= 16 \\ \mathrm{b}_{ 14 } &= 23 \\ \mathrm{b}_{ 16 } &= 33 \\ \mathrm{b}_{ 18 } &= 46 \\ \mathrm{b}_{ 20 } &= 63 \\ \mathrm{b}_{ 22 } &= 83 \\ \mathrm{b}_{ 24 } &= 110 \\ \mathrm{b}_{ 26 } &= 142 \\ \mathrm{b}_{ 28 } &= 180 \\ \mathrm{b}_{ 30 } &= 227 \\ \mathrm{b}_{ 32 } &= 282 \\ \mathrm{b}_{ 34 } &= 345 \\ \mathrm{b}_{ 36 } &= 419 \\ \mathrm{b}_{ 38 } &= 505 \\ \mathrm{b}_{ 40 } &= 600 \\ \mathrm{b}_{ 42 } &= 709 \\ \mathrm{b}_{ 44 } &= 830 \\ \mathrm{b}_{ 46 } &= 964 \\ \mathrm{b}_{ 48 } &= 1110 \\ \mathrm{b}_{ 50 } &= 1273 \\ \mathrm{b}_{ 52 } &= 1445 \\ \mathrm{b}_{ 54 } &= 1632 \\ \mathrm{b}_{ 56 } &= 1831 \\ \mathrm{b}_{ 58 } &= 2043 \\ \mathrm{b}_{ 60 } &= 2263 \\ \mathrm{b}_{ 62 } &= 2497 \\ \mathrm{b}_{ 64 } &= 2736 \\ \mathrm{b}_{ 66 } &= 2983 \\ \mathrm{b}_{ 68 } &= 3236 \\ \mathrm{b}_{ 70 } &= 3494 \\ \mathrm{b}_{ 72 } &= 3750 \\ \mathrm{b}_{ 74 } &= 4009 \\ \mathrm{b}_{ 76 } &= 4265 \\ \mathrm{b}_{ 78 } &= 4514 \\ \mathrm{b}_{ 80 } &= 4758 \\ \mathrm{b}_{ 82 } &= 4994 \\ \mathrm{b}_{ 84 } &= 5216 \\ \mathrm{b}_{ 86 } &= 5424 \\ \mathrm{b}_{ 88 } &= 5620 \\ \mathrm{b}_{ 90 } &= 5794 \\ \mathrm{b}_{ 92 } &= 5951 \\ \mathrm{b}_{ 94 } &= 6087 \\ \mathrm{b}_{ 96 } &= 6200 \\ \mathrm{b}_{ 98 } &= 6286 \\ \mathrm{b}_{ 100 } &= 6354 \\ \mathrm{b}_{ 102 } &= 6391 \\ \mathrm{b}_{ 104 } &= 6404 \\ \mathrm{b}_{ 106 } &= 6391 \\ \mathrm{b}_{ 108 } &= 6354 \\ \mathrm{b}_{ 110 } &= 6286 \\ \mathrm{b}_{ 112 } &= 6200 \\ \mathrm{b}_{ 114 } &= 6087 \\ \mathrm{b}_{ 116 } &= 5951 \\ \mathrm{b}_{ 118 } &= 5794 \\ \mathrm{b}_{ 120 } &= 5620 \\ \mathrm{b}_{ 122 } &= 5424 \\ \mathrm{b}_{ 124 } &= 5216 \\ \mathrm{b}_{ 126 } &= 4994 \\ \mathrm{b}_{ 128 } &= 4758 \\ \mathrm{b}_{ 130 } &= 4514 \\ \mathrm{b}_{ 132 } &= 4265 \\ \mathrm{b}_{ 134 } &= 4009 \\ \mathrm{b}_{ 136 } &= 3750 \\ \mathrm{b}_{ 138 } &= 3494 \\ \mathrm{b}_{ 140 } &= 3236 \\ \mathrm{b}_{ 142 } &= 2983 \\ \mathrm{b}_{ 144 } &= 2736 \\ \mathrm{b}_{ 146 } &= 2497 \\ \mathrm{b}_{ 148 } &= 2263 \\ \mathrm{b}_{ 150 } &= 2043 \\ \mathrm{b}_{ 152 } &= 1831 \\ \mathrm{b}_{ 154 } &= 1632 \\ \mathrm{b}_{ 156 } &= 1445 \\ \mathrm{b}_{ 158 } &= 1273 \\ \mathrm{b}_{ 160 } &= 1110 \\ \mathrm{b}_{ 162 } &= 964 \\ \mathrm{b}_{ 164 } &= 830 \\ \mathrm{b}_{ 166 } &= 709 \\ \mathrm{b}_{ 168 } &= 600 \\ \mathrm{b}_{ 170 } &= 505 \\ \mathrm{b}_{ 172 } &= 419 \\ \mathrm{b}_{ 174 } &= 345 \\ \mathrm{b}_{ 176 } &= 282 \\ \mathrm{b}_{ 178 } &= 227 \\ \mathrm{b}_{ 180 } &= 180 \\ \mathrm{b}_{ 182 } &= 142 \\ \mathrm{b}_{ 184 } &= 110 \\ \mathrm{b}_{ 186 } &= 83 \\ \mathrm{b}_{ 188 } &= 63 \\ \mathrm{b}_{ 190 } &= 46 \\ \mathrm{b}_{ 192 } &= 33 \\ \mathrm{b}_{ 194 } &= 23 \\ \mathrm{b}_{ 196 } &= 16 \\ \mathrm{b}_{ 198 } &= 10 \\ \mathrm{b}_{ 200 } &= 7 \\ \mathrm{b}_{ 202 } &= 4 \\ \mathrm{b}_{ 204 } &= 2 \\ \mathrm{b}_{ 206 } &= 1 \\ \mathrm{b}_{ 208 } &= 1 \end{align*}
Basic information
dimension
104
index
11
Euler characteristic
241920
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 8 } = 7$, $\mathrm{b}_{ 10 } = 10$, $\mathrm{b}_{ 12 } = 16$, $\mathrm{b}_{ 14 } = 23$, $\mathrm{b}_{ 16 } = 33$, $\mathrm{b}_{ 18 } = 46$, $\mathrm{b}_{ 20 } = 63$, $\mathrm{b}_{ 22 } = 83$, $\mathrm{b}_{ 24 } = 110$, $\mathrm{b}_{ 26 } = 142$, $\mathrm{b}_{ 28 } = 180$, $\mathrm{b}_{ 30 } = 227$, $\mathrm{b}_{ 32 } = 282$, $\mathrm{b}_{ 34 } = 345$, $\mathrm{b}_{ 36 } = 419$, $\mathrm{b}_{ 38 } = 505$, $\mathrm{b}_{ 40 } = 600$, $\mathrm{b}_{ 42 } = 709$, $\mathrm{b}_{ 44 } = 830$, $\mathrm{b}_{ 46 } = 964$, $\mathrm{b}_{ 48 } = 1110$, $\mathrm{b}_{ 50 } = 1273$, $\mathrm{b}_{ 52 } = 1445$, $\mathrm{b}_{ 54 } = 1632$, $\mathrm{b}_{ 56 } = 1831$, $\mathrm{b}_{ 58 } = 2043$, $\mathrm{b}_{ 60 } = 2263$, $\mathrm{b}_{ 62 } = 2497$, $\mathrm{b}_{ 64 } = 2736$, $\mathrm{b}_{ 66 } = 2983$, $\mathrm{b}_{ 68 } = 3236$, $\mathrm{b}_{ 70 } = 3494$, $\mathrm{b}_{ 72 } = 3750$, $\mathrm{b}_{ 74 } = 4009$, $\mathrm{b}_{ 76 } = 4265$, $\mathrm{b}_{ 78 } = 4514$, $\mathrm{b}_{ 80 } = 4758$, $\mathrm{b}_{ 82 } = 4994$, $\mathrm{b}_{ 84 } = 5216$, $\mathrm{b}_{ 86 } = 5424$, $\mathrm{b}_{ 88 } = 5620$, $\mathrm{b}_{ 90 } = 5794$, $\mathrm{b}_{ 92 } = 5951$, $\mathrm{b}_{ 94 } = 6087$, $\mathrm{b}_{ 96 } = 6200$, $\mathrm{b}_{ 98 } = 6286$, $\mathrm{b}_{ 100 } = 6354$, $\mathrm{b}_{ 102 } = 6391$, $\mathrm{b}_{ 104 } = 6404$, $\mathrm{b}_{ 106 } = 6391$, $\mathrm{b}_{ 108 } = 6354$, $\mathrm{b}_{ 110 } = 6286$, $\mathrm{b}_{ 112 } = 6200$, $\mathrm{b}_{ 114 } = 6087$, $\mathrm{b}_{ 116 } = 5951$, $\mathrm{b}_{ 118 } = 5794$, $\mathrm{b}_{ 120 } = 5620$, $\mathrm{b}_{ 122 } = 5424$, $\mathrm{b}_{ 124 } = 5216$, $\mathrm{b}_{ 126 } = 4994$, $\mathrm{b}_{ 128 } = 4758$, $\mathrm{b}_{ 130 } = 4514$, $\mathrm{b}_{ 132 } = 4265$, $\mathrm{b}_{ 134 } = 4009$, $\mathrm{b}_{ 136 } = 3750$, $\mathrm{b}_{ 138 } = 3494$, $\mathrm{b}_{ 140 } = 3236$, $\mathrm{b}_{ 142 } = 2983$, $\mathrm{b}_{ 144 } = 2736$, $\mathrm{b}_{ 146 } = 2497$, $\mathrm{b}_{ 148 } = 2263$, $\mathrm{b}_{ 150 } = 2043$, $\mathrm{b}_{ 152 } = 1831$, $\mathrm{b}_{ 154 } = 1632$, $\mathrm{b}_{ 156 } = 1445$, $\mathrm{b}_{ 158 } = 1273$, $\mathrm{b}_{ 160 } = 1110$, $\mathrm{b}_{ 162 } = 964$, $\mathrm{b}_{ 164 } = 830$, $\mathrm{b}_{ 166 } = 709$, $\mathrm{b}_{ 168 } = 600$, $\mathrm{b}_{ 170 } = 505$, $\mathrm{b}_{ 172 } = 419$, $\mathrm{b}_{ 174 } = 345$, $\mathrm{b}_{ 176 } = 282$, $\mathrm{b}_{ 178 } = 227$, $\mathrm{b}_{ 180 } = 180$, $\mathrm{b}_{ 182 } = 142$, $\mathrm{b}_{ 184 } = 110$, $\mathrm{b}_{ 186 } = 83$, $\mathrm{b}_{ 188 } = 63$, $\mathrm{b}_{ 190 } = 46$, $\mathrm{b}_{ 192 } = 33$, $\mathrm{b}_{ 194 } = 23$, $\mathrm{b}_{ 196 } = 16$, $\mathrm{b}_{ 198 } = 10$, $\mathrm{b}_{ 200 } = 7$, $\mathrm{b}_{ 202 } = 4$, $\mathrm{b}_{ 204 } = 2$, $\mathrm{b}_{ 206 } = 1$, $\mathrm{b}_{ 208 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{5})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{5})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{5})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{5}\hookrightarrow\mathbb{P}^{ 146325269 }$

degree
7048739117760336797977983562340287515545642544037932809921794017244193619051628462080000000
Hilbert series
1, 146325270, 923737544513280, 673728676102290528000, 98438879670601838707170594, 4106385980139520203749853150096, 62498933585649921714653231155223000, 414704422648739536115316324441710625000, 1371539926800853261014616935058659187148625, 2507352927826542920486079415333658912848480704, 2749498823341566890746193988952937699308220710912, 1931423115372830253045999901312316264810501111808000, 917102776760837657362195094150087719150655485796329920, 307746837536532018461265741712928892435469104856657389105, 75750702874144487920130480174968837836431569963502722253172, 14115358495717738210145588803129381124237343817989284742106656, 2045521966538116206277491063577828391170971802121392621347518720, 235934547741648649266381259941745387764650057447152216376534560500, 22099642548451339920047614076156522323900895798881320327668220000000, 1710830893273559080266142126405131916879071362470422370148437500000000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{5})$. 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