Grassmannian.info

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

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, ...

The projective dual is a hypersurface.

codimension

1

degree

12675062580328896449783563398379832053473320339262459713279209837335366870998230827544320

self-dual

no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Tevelev, Theorems 7.1–7.2 (arXiv version)] All Grassmannians.

Full exceptional collections in $\mathbf D^{\mathrm b}(\mathrm{E}_{8}/\mathrm{P}_{5})$:

• Samokhin–van der Kallen (2024)

small

not generically semisimple [Galkin–Leung–Li–Xiong, Theorem 1.2]

Graded dimensions modulo the index $11$: $\widetilde b_{ 1 }=21993>21992=\widetilde b_{ 0 }$, where $0\equiv 0\cdot 1\pmod{ 11 }$.

big

generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

No Lefschetz decomposition recorded.

No HP-dual model recorded.

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

1. 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.
2. 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.
3. Sergey Galkin; Naichung Conan Leung; Changzheng Li; Rui Xiong; appendix by Pieter Belmans and Markus Reineke. A-D-E diagrams, Hodge–Tate hyperplane sections and semisimple quantum cohomology. Preprint (2025). arXiv:2509.01101.
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.