\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 E6/P5

There exist other realisations of this Grassmannian:
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 } &= 6 \\ \mathrm{b}_{ 12 } &= 8 \\ \mathrm{b}_{ 14 } &= 10 \\ \mathrm{b}_{ 16 } &= 12 \\ \mathrm{b}_{ 18 } &= 13 \\ \mathrm{b}_{ 20 } &= 15 \\ \mathrm{b}_{ 22 } &= 16 \\ \mathrm{b}_{ 24 } &= 16 \\ \mathrm{b}_{ 26 } &= 16 \\ \mathrm{b}_{ 28 } &= 16 \\ \mathrm{b}_{ 30 } &= 15 \\ \mathrm{b}_{ 32 } &= 13 \\ \mathrm{b}_{ 34 } &= 12 \\ \mathrm{b}_{ 36 } &= 10 \\ \mathrm{b}_{ 38 } &= 8 \\ \mathrm{b}_{ 40 } &= 6 \\ \mathrm{b}_{ 42 } &= 5 \\ \mathrm{b}_{ 44 } &= 3 \\ \mathrm{b}_{ 46 } &= 2 \\ \mathrm{b}_{ 48 } &= 1 \\ \mathrm{b}_{ 50 } &= 1 \end{align*}
Basic information
dimension
25
index
9
Euler characteristic
216
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 5$, $\mathrm{b}_{ 10 } = 6$, $\mathrm{b}_{ 12 } = 8$, $\mathrm{b}_{ 14 } = 10$, $\mathrm{b}_{ 16 } = 12$, $\mathrm{b}_{ 18 } = 13$, $\mathrm{b}_{ 20 } = 15$, $\mathrm{b}_{ 22 } = 16$, $\mathrm{b}_{ 24 } = 16$, $\mathrm{b}_{ 26 } = 16$, $\mathrm{b}_{ 28 } = 16$, $\mathrm{b}_{ 30 } = 15$, $\mathrm{b}_{ 32 } = 13$, $\mathrm{b}_{ 34 } = 12$, $\mathrm{b}_{ 36 } = 10$, $\mathrm{b}_{ 38 } = 8$, $\mathrm{b}_{ 40 } = 6$, $\mathrm{b}_{ 42 } = 5$, $\mathrm{b}_{ 44 } = 3$, $\mathrm{b}_{ 46 } = 2$, $\mathrm{b}_{ 48 } = 1$, $\mathrm{b}_{ 50 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{6}/\mathrm{P}_{5})$
adjoint group of type $\mathrm{E}_{ 6 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{6}/\mathrm{P}_{5})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{6}/\mathrm{P}_{5})$
78
Projective geometry
minimal embedding

$\mathrm{E}_{6}/\mathrm{P}_{5}\hookrightarrow\mathbb{P}^{ 350 }$

degree
6119118720
Hilbert series
1, 351, 34398, 1559376, 41442192, 740779182, 9731535957, 99876290085, 836979819390, 5919754500660, 36248385923568, 196056157459968, 951851777730048, 4202954077177584, 17062684674908652, 64266967368492174, 226311318775941399, 749969516734275915, 2352031814236869050, 7014911034217086000, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
17796720
self-dual
no

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

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\mathrm{E}_{6}/\mathrm{P}_{5})$:

Quantum cohomology
small
generically semisimple [Belmans (quantum spectra)]
big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{6}/\mathrm{P}_{5})$:

Homological projective duality

No Lefschetz decomposition recorded.

No HP-dual model recorded.

$k$-Fanoness

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

References
  1. Pieter Belmans. An update for Grassmannian.info: quantum spectra for horospherical varieties. Research note, 24 February 2021.
  2. 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.
  3. 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.
  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.