\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/P6

Betti numbers
\begin{align*} \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 } &= 14 \\ \mathrm{b}_{ 16 } &= 20 \\ \mathrm{b}_{ 18 } &= 26 \\ \mathrm{b}_{ 20 } &= 35 \\ \mathrm{b}_{ 22 } &= 44 \\ \mathrm{b}_{ 24 } &= 57 \\ \mathrm{b}_{ 26 } &= 70 \\ \mathrm{b}_{ 28 } &= 87 \\ \mathrm{b}_{ 30 } &= 106 \\ \mathrm{b}_{ 32 } &= 129 \\ \mathrm{b}_{ 34 } &= 153 \\ \mathrm{b}_{ 36 } &= 182 \\ \mathrm{b}_{ 38 } &= 213 \\ \mathrm{b}_{ 40 } &= 248 \\ \mathrm{b}_{ 42 } &= 287 \\ \mathrm{b}_{ 44 } &= 329 \\ \mathrm{b}_{ 46 } &= 374 \\ \mathrm{b}_{ 48 } &= 422 \\ \mathrm{b}_{ 50 } &= 475 \\ \mathrm{b}_{ 52 } &= 529 \\ \mathrm{b}_{ 54 } &= 588 \\ \mathrm{b}_{ 56 } &= 647 \\ \mathrm{b}_{ 58 } &= 710 \\ \mathrm{b}_{ 60 } &= 772 \\ \mathrm{b}_{ 62 } &= 840 \\ \mathrm{b}_{ 64 } &= 904 \\ \mathrm{b}_{ 66 } &= 972 \\ \mathrm{b}_{ 68 } &= 1036 \\ \mathrm{b}_{ 70 } &= 1103 \\ \mathrm{b}_{ 72 } &= 1164 \\ \mathrm{b}_{ 74 } &= 1229 \\ \mathrm{b}_{ 76 } &= 1286 \\ \mathrm{b}_{ 78 } &= 1343 \\ \mathrm{b}_{ 80 } &= 1393 \\ \mathrm{b}_{ 82 } &= 1443 \\ \mathrm{b}_{ 84 } &= 1484 \\ \mathrm{b}_{ 86 } &= 1524 \\ \mathrm{b}_{ 88 } &= 1555 \\ \mathrm{b}_{ 90 } &= 1581 \\ \mathrm{b}_{ 92 } &= 1600 \\ \mathrm{b}_{ 94 } &= 1615 \\ \mathrm{b}_{ 96 } &= 1621 \\ \mathrm{b}_{ 98 } &= 1621 \\ \mathrm{b}_{ 100 } &= 1615 \\ \mathrm{b}_{ 102 } &= 1600 \\ \mathrm{b}_{ 104 } &= 1581 \\ \mathrm{b}_{ 106 } &= 1555 \\ \mathrm{b}_{ 108 } &= 1524 \\ \mathrm{b}_{ 110 } &= 1484 \\ \mathrm{b}_{ 112 } &= 1443 \\ \mathrm{b}_{ 114 } &= 1393 \\ \mathrm{b}_{ 116 } &= 1343 \\ \mathrm{b}_{ 118 } &= 1286 \\ \mathrm{b}_{ 120 } &= 1229 \\ \mathrm{b}_{ 122 } &= 1164 \\ \mathrm{b}_{ 124 } &= 1103 \\ \mathrm{b}_{ 126 } &= 1036 \\ \mathrm{b}_{ 128 } &= 972 \\ \mathrm{b}_{ 130 } &= 904 \\ \mathrm{b}_{ 132 } &= 840 \\ \mathrm{b}_{ 134 } &= 772 \\ \mathrm{b}_{ 136 } &= 710 \\ \mathrm{b}_{ 138 } &= 647 \\ \mathrm{b}_{ 140 } &= 588 \\ \mathrm{b}_{ 142 } &= 529 \\ \mathrm{b}_{ 144 } &= 475 \\ \mathrm{b}_{ 146 } &= 422 \\ \mathrm{b}_{ 148 } &= 374 \\ \mathrm{b}_{ 150 } &= 329 \\ \mathrm{b}_{ 152 } &= 287 \\ \mathrm{b}_{ 154 } &= 248 \\ \mathrm{b}_{ 156 } &= 213 \\ \mathrm{b}_{ 158 } &= 182 \\ \mathrm{b}_{ 160 } &= 153 \\ \mathrm{b}_{ 162 } &= 129 \\ \mathrm{b}_{ 164 } &= 106 \\ \mathrm{b}_{ 166 } &= 87 \\ \mathrm{b}_{ 168 } &= 70 \\ \mathrm{b}_{ 170 } &= 57 \\ \mathrm{b}_{ 172 } &= 44 \\ \mathrm{b}_{ 174 } &= 35 \\ \mathrm{b}_{ 176 } &= 26 \\ \mathrm{b}_{ 178 } &= 20 \\ \mathrm{b}_{ 180 } &= 14 \\ \mathrm{b}_{ 182 } &= 11 \\ \mathrm{b}_{ 184 } &= 7 \\ \mathrm{b}_{ 186 } &= 5 \\ \mathrm{b}_{ 188 } &= 3 \\ \mathrm{b}_{ 190 } &= 2 \\ \mathrm{b}_{ 192 } &= 1 \\ \mathrm{b}_{ 194 } &= 1 \end{align*}
Basic information
dimension
97
index
14
Euler characteristic
60480
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 7$, $\mathrm{b}_{ 12 } = 11$, $\mathrm{b}_{ 14 } = 14$, $\mathrm{b}_{ 16 } = 20$, $\mathrm{b}_{ 18 } = 26$, $\mathrm{b}_{ 20 } = 35$, $\mathrm{b}_{ 22 } = 44$, $\mathrm{b}_{ 24 } = 57$, $\mathrm{b}_{ 26 } = 70$, $\mathrm{b}_{ 28 } = 87$, $\mathrm{b}_{ 30 } = 106$, $\mathrm{b}_{ 32 } = 129$, $\mathrm{b}_{ 34 } = 153$, $\mathrm{b}_{ 36 } = 182$, $\mathrm{b}_{ 38 } = 213$, $\mathrm{b}_{ 40 } = 248$, $\mathrm{b}_{ 42 } = 287$, $\mathrm{b}_{ 44 } = 329$, $\mathrm{b}_{ 46 } = 374$, $\mathrm{b}_{ 48 } = 422$, $\mathrm{b}_{ 50 } = 475$, $\mathrm{b}_{ 52 } = 529$, $\mathrm{b}_{ 54 } = 588$, $\mathrm{b}_{ 56 } = 647$, $\mathrm{b}_{ 58 } = 710$, $\mathrm{b}_{ 60 } = 772$, $\mathrm{b}_{ 62 } = 840$, $\mathrm{b}_{ 64 } = 904$, $\mathrm{b}_{ 66 } = 972$, $\mathrm{b}_{ 68 } = 1036$, $\mathrm{b}_{ 70 } = 1103$, $\mathrm{b}_{ 72 } = 1164$, $\mathrm{b}_{ 74 } = 1229$, $\mathrm{b}_{ 76 } = 1286$, $\mathrm{b}_{ 78 } = 1343$, $\mathrm{b}_{ 80 } = 1393$, $\mathrm{b}_{ 82 } = 1443$, $\mathrm{b}_{ 84 } = 1484$, $\mathrm{b}_{ 86 } = 1524$, $\mathrm{b}_{ 88 } = 1555$, $\mathrm{b}_{ 90 } = 1581$, $\mathrm{b}_{ 92 } = 1600$, $\mathrm{b}_{ 94 } = 1615$, $\mathrm{b}_{ 96 } = 1621$, $\mathrm{b}_{ 98 } = 1621$, $\mathrm{b}_{ 100 } = 1615$, $\mathrm{b}_{ 102 } = 1600$, $\mathrm{b}_{ 104 } = 1581$, $\mathrm{b}_{ 106 } = 1555$, $\mathrm{b}_{ 108 } = 1524$, $\mathrm{b}_{ 110 } = 1484$, $\mathrm{b}_{ 112 } = 1443$, $\mathrm{b}_{ 114 } = 1393$, $\mathrm{b}_{ 116 } = 1343$, $\mathrm{b}_{ 118 } = 1286$, $\mathrm{b}_{ 120 } = 1229$, $\mathrm{b}_{ 122 } = 1164$, $\mathrm{b}_{ 124 } = 1103$, $\mathrm{b}_{ 126 } = 1036$, $\mathrm{b}_{ 128 } = 972$, $\mathrm{b}_{ 130 } = 904$, $\mathrm{b}_{ 132 } = 840$, $\mathrm{b}_{ 134 } = 772$, $\mathrm{b}_{ 136 } = 710$, $\mathrm{b}_{ 138 } = 647$, $\mathrm{b}_{ 140 } = 588$, $\mathrm{b}_{ 142 } = 529$, $\mathrm{b}_{ 144 } = 475$, $\mathrm{b}_{ 146 } = 422$, $\mathrm{b}_{ 148 } = 374$, $\mathrm{b}_{ 150 } = 329$, $\mathrm{b}_{ 152 } = 287$, $\mathrm{b}_{ 154 } = 248$, $\mathrm{b}_{ 156 } = 213$, $\mathrm{b}_{ 158 } = 182$, $\mathrm{b}_{ 160 } = 153$, $\mathrm{b}_{ 162 } = 129$, $\mathrm{b}_{ 164 } = 106$, $\mathrm{b}_{ 166 } = 87$, $\mathrm{b}_{ 168 } = 70$, $\mathrm{b}_{ 170 } = 57$, $\mathrm{b}_{ 172 } = 44$, $\mathrm{b}_{ 174 } = 35$, $\mathrm{b}_{ 176 } = 26$, $\mathrm{b}_{ 178 } = 20$, $\mathrm{b}_{ 180 } = 14$, $\mathrm{b}_{ 182 } = 11$, $\mathrm{b}_{ 184 } = 7$, $\mathrm{b}_{ 186 } = 5$, $\mathrm{b}_{ 188 } = 3$, $\mathrm{b}_{ 190 } = 2$, $\mathrm{b}_{ 192 } = 1$, $\mathrm{b}_{ 194 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{6})$
adjoint group of type $\mathrm{E}_{ 8 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{8}/\mathrm{P}_{6})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{8}/\mathrm{P}_{6})$
248
Projective geometry
minimal embedding

$\mathrm{E}_{8}/\mathrm{P}_{6}\hookrightarrow\mathbb{P}^{ 2450239 }$

degree
4000401838497964449851582384149404943358006439290583996021880689524736000
Hilbert series
1, 2450240, 627099023250, 33834524434480000, 563050913077505352500, 3692223011493517613112000, 11322104972068767946410271125, 18412978008469718999941611840000, 17479127232407059372193470597440000, 10440078307076271004292040576008192000, 4166115165359402681083208580614639613440, 1166273891835999947812449977021140612557312, 238458629917484093396266298568333681210473875, 36829875977795892871805413738643649224714640000, 4421076436213296810675693671516812004811177230000, 422617463879756942550163766818677231159619320768000, 32850163044654279981338110790301110753012338509562500, 2114315457670604076601980353685564460776251709375000000, 114476208547168269004786488917296625254436369659423828125, 5286990930545438070228765179864284377184301069353125000000, ...
Exceptional collections

No full exceptional collection is known for $\mathbf{D}^{\mathrm{b}}(\mathrm{E}_{8}/\mathrm{P}_{6})$. Will you be the first to construct one? Let us know if you do!

Quantum cohomology

The small quantum cohomology is not known to be (non-)semisimple.

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

Homological projective duality