\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 E7/P5

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 4 \\ \mathrm{b}_{ 8 } &= 6 \\ \mathrm{b}_{ 10 } &= 9 \\ \mathrm{b}_{ 12 } &= 13 \\ \mathrm{b}_{ 14 } &= 18 \\ \mathrm{b}_{ 16 } &= 24 \\ \mathrm{b}_{ 18 } &= 32 \\ \mathrm{b}_{ 20 } &= 40 \\ \mathrm{b}_{ 22 } &= 50 \\ \mathrm{b}_{ 24 } &= 61 \\ \mathrm{b}_{ 26 } &= 73 \\ \mathrm{b}_{ 28 } &= 85 \\ \mathrm{b}_{ 30 } &= 99 \\ \mathrm{b}_{ 32 } &= 112 \\ \mathrm{b}_{ 34 } &= 125 \\ \mathrm{b}_{ 36 } &= 138 \\ \mathrm{b}_{ 38 } &= 150 \\ \mathrm{b}_{ 40 } &= 161 \\ \mathrm{b}_{ 42 } &= 170 \\ \mathrm{b}_{ 44 } &= 178 \\ \mathrm{b}_{ 46 } &= 183 \\ \mathrm{b}_{ 48 } &= 187 \\ \mathrm{b}_{ 50 } &= 188 \\ \mathrm{b}_{ 52 } &= 187 \\ \mathrm{b}_{ 54 } &= 183 \\ \mathrm{b}_{ 56 } &= 178 \\ \mathrm{b}_{ 58 } &= 170 \\ \mathrm{b}_{ 60 } &= 161 \\ \mathrm{b}_{ 62 } &= 150 \\ \mathrm{b}_{ 64 } &= 138 \\ \mathrm{b}_{ 66 } &= 125 \\ \mathrm{b}_{ 68 } &= 112 \\ \mathrm{b}_{ 70 } &= 99 \\ \mathrm{b}_{ 72 } &= 85 \\ \mathrm{b}_{ 74 } &= 73 \\ \mathrm{b}_{ 76 } &= 61 \\ \mathrm{b}_{ 78 } &= 50 \\ \mathrm{b}_{ 80 } &= 40 \\ \mathrm{b}_{ 82 } &= 32 \\ \mathrm{b}_{ 84 } &= 24 \\ \mathrm{b}_{ 86 } &= 18 \\ \mathrm{b}_{ 88 } &= 13 \\ \mathrm{b}_{ 90 } &= 9 \\ \mathrm{b}_{ 92 } &= 6 \\ \mathrm{b}_{ 94 } &= 4 \\ \mathrm{b}_{ 96 } &= 2 \\ \mathrm{b}_{ 98 } &= 1 \\ \mathrm{b}_{ 100 } &= 1 \end{align*}
Basic information
dimension
50
index
10
Euler characteristic
4032
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 4$, $\mathrm{b}_{ 8 } = 6$, $\mathrm{b}_{ 10 } = 9$, $\mathrm{b}_{ 12 } = 13$, $\mathrm{b}_{ 14 } = 18$, $\mathrm{b}_{ 16 } = 24$, $\mathrm{b}_{ 18 } = 32$, $\mathrm{b}_{ 20 } = 40$, $\mathrm{b}_{ 22 } = 50$, $\mathrm{b}_{ 24 } = 61$, $\mathrm{b}_{ 26 } = 73$, $\mathrm{b}_{ 28 } = 85$, $\mathrm{b}_{ 30 } = 99$, $\mathrm{b}_{ 32 } = 112$, $\mathrm{b}_{ 34 } = 125$, $\mathrm{b}_{ 36 } = 138$, $\mathrm{b}_{ 38 } = 150$, $\mathrm{b}_{ 40 } = 161$, $\mathrm{b}_{ 42 } = 170$, $\mathrm{b}_{ 44 } = 178$, $\mathrm{b}_{ 46 } = 183$, $\mathrm{b}_{ 48 } = 187$, $\mathrm{b}_{ 50 } = 188$, $\mathrm{b}_{ 52 } = 187$, $\mathrm{b}_{ 54 } = 183$, $\mathrm{b}_{ 56 } = 178$, $\mathrm{b}_{ 58 } = 170$, $\mathrm{b}_{ 60 } = 161$, $\mathrm{b}_{ 62 } = 150$, $\mathrm{b}_{ 64 } = 138$, $\mathrm{b}_{ 66 } = 125$, $\mathrm{b}_{ 68 } = 112$, $\mathrm{b}_{ 70 } = 99$, $\mathrm{b}_{ 72 } = 85$, $\mathrm{b}_{ 74 } = 73$, $\mathrm{b}_{ 76 } = 61$, $\mathrm{b}_{ 78 } = 50$, $\mathrm{b}_{ 80 } = 40$, $\mathrm{b}_{ 82 } = 32$, $\mathrm{b}_{ 84 } = 24$, $\mathrm{b}_{ 86 } = 18$, $\mathrm{b}_{ 88 } = 13$, $\mathrm{b}_{ 90 } = 9$, $\mathrm{b}_{ 92 } = 6$, $\mathrm{b}_{ 94 } = 4$, $\mathrm{b}_{ 96 } = 2$, $\mathrm{b}_{ 98 } = 1$, $\mathrm{b}_{ 100 } = 1$
$\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{5})$
adjoint group of type $\mathrm{E}_{ 7 }$
$\pi_0\mathrm{Aut}(\mathrm{E}_{7}/\mathrm{P}_{5})$
$1$
$\dim\mathrm{Aut}^0(\mathrm{E}_{7}/\mathrm{P}_{5})$
133
Projective geometry
minimal embedding

$\mathrm{E}_{7}/\mathrm{P}_{5}\hookrightarrow\mathbb{P}^{ 27663 }$

degree
7189094617369512864411864268800
Hilbert series
1, 27664, 109120648, 122211210240, 55316934276120, 12558300006045024, 1652084830739933590, 139491450390045039000, 8150914397472701604750, 349067632878637409842560, 11457227739469333306727424, 298660275350729176734760960, 6363906479714806736751313920, 113502798476673004420339183488, 1728171187811885553111503151894, 22838477706875454786731446453576, 265685045394342480356361068851540, 2753789609279788667104987042619200, 25697281478091630912981145307715000, 217858949465800585414942950480000000, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
16605160862605167542808003744
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 $\mathbf D^{\mathrm b}(\mathrm{E}_{7}/\mathrm{P}_{5})$:

Quantum cohomology
small
generically semisimple [Holmes–Muratore, Theorems 1.2 and 6.2; Remark 6.3]

Semisimple at $q=1$; $c_1\star$ has the eigenvalue $0$ with algebraic multiplicity 12.

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

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\mathrm{E}_{7}/\mathrm{P}_{5})$:

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. 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. Daniel Holmes; Giosuè Muratore. Effective Gromov–Witten computations for GKM spaces and a semisimple application. Preprint (2026). arXiv:2610.37273.
  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.