Grassmannian.info

dimension

42

index

13

Euler characteristic

756

Betti numbers

$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 11$, $\mathrm{b}_{ 18 } = 13$, $\mathrm{b}_{ 20 } = 16$, $\mathrm{b}_{ 22 } = 18$, $\mathrm{b}_{ 24 } = 21$, $\mathrm{b}_{ 26 } = 24$, $\mathrm{b}_{ 28 } = 26$, $\mathrm{b}_{ 30 } = 29$, $\mathrm{b}_{ 32 } = 31$, $\mathrm{b}_{ 34 } = 34$, $\mathrm{b}_{ 36 } = 34$, $\mathrm{b}_{ 38 } = 36$, $\mathrm{b}_{ 40 } = 36$, $\mathrm{b}_{ 42 } = 38$, $\mathrm{b}_{ 44 } = 36$, $\mathrm{b}_{ 46 } = 36$, $\mathrm{b}_{ 48 } = 34$, $\mathrm{b}_{ 50 } = 34$, $\mathrm{b}_{ 52 } = 31$, $\mathrm{b}_{ 54 } = 29$, $\mathrm{b}_{ 56 } = 26$, $\mathrm{b}_{ 58 } = 24$, $\mathrm{b}_{ 60 } = 21$, $\mathrm{b}_{ 62 } = 18$, $\mathrm{b}_{ 64 } = 16$, $\mathrm{b}_{ 66 } = 13$, $\mathrm{b}_{ 68 } = 11$, $\mathrm{b}_{ 70 } = 8$, $\mathrm{b}_{ 72 } = 7$, $\mathrm{b}_{ 74 } = 5$, $\mathrm{b}_{ 76 } = 4$, $\mathrm{b}_{ 78 } = 2$, $\mathrm{b}_{ 80 } = 2$, $\mathrm{b}_{ 82 } = 1$, $\mathrm{b}_{ 84 } = 1$

$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{6})$

adjoint group of type $\mathrm{E}_{ 7 }$

$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{6})$

$1$

$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{6})$

133

minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{6}\hookrightarrow\mathbb{P}^{ 1538 }$

degree

6457628866020433920

Hilbert series

1, 1539, 617253, 105489615, 9743909175, 561104814270, 22155294211050, 641798777779380, 14341812253735200, 256924158460700640, 3803151504372077088, 47661484991340720864, 515805481647912249276, 4900345026363722587050, 41434653012551675362750, 315469600749446851604325, 2184413433392140346347875, 13874772955031952078740625, 81446133440497985177484375, 444734992730941305619078125, ...

The projective dual is a hypersurface.

codimension

1

degree

434639607154428

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 $\mathbf D^{\mathrm b}(\mathrm{E}_{7}/\mathrm{P}_{6})$:

• Samokhin–van der Kallen (2024)

small

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

Graded dimensions modulo the index $13$: $\widetilde b_{ 1 }=58>57=\widetilde b_{ 5 }$, where $5\equiv 5\cdot 1\pmod{ 13 }$.

big

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

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{7}/\mathrm{P}_{6})$:

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.