\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.

Symplectic Grassmannian $\SGr(3,12)$

Y Z Y Z Y Z Y Z Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 10 } &= 5 \\ \mathrm{b}_{ 12 } &= 7 \\ \mathrm{b}_{ 14 } &= 8 \\ \mathrm{b}_{ 16 } &= 9 \\ \mathrm{b}_{ 18 } &= 11 \\ \mathrm{b}_{ 20 } &= 11 \\ \mathrm{b}_{ 22 } &= 12 \\ \mathrm{b}_{ 24 } &= 12 \\ \mathrm{b}_{ 26 } &= 12 \\ \mathrm{b}_{ 28 } &= 11 \\ \mathrm{b}_{ 30 } &= 11 \\ \mathrm{b}_{ 32 } &= 9 \\ \mathrm{b}_{ 34 } &= 8 \\ \mathrm{b}_{ 36 } &= 7 \\ \mathrm{b}_{ 38 } &= 5 \\ \mathrm{b}_{ 40 } &= 4 \\ \mathrm{b}_{ 42 } &= 3 \\ \mathrm{b}_{ 44 } &= 2 \\ \mathrm{b}_{ 46 } &= 1 \\ \mathrm{b}_{ 48 } &= 1 \end{align*}
Basic information
dimension
24
index
10
Euler characteristic
160
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 5$, $\mathrm{b}_{ 12 } = 7$, $\mathrm{b}_{ 14 } = 8$, $\mathrm{b}_{ 16 } = 9$, $\mathrm{b}_{ 18 } = 11$, $\mathrm{b}_{ 20 } = 11$, $\mathrm{b}_{ 22 } = 12$, $\mathrm{b}_{ 24 } = 12$, $\mathrm{b}_{ 26 } = 12$, $\mathrm{b}_{ 28 } = 11$, $\mathrm{b}_{ 30 } = 11$, $\mathrm{b}_{ 32 } = 9$, $\mathrm{b}_{ 34 } = 8$, $\mathrm{b}_{ 36 } = 7$, $\mathrm{b}_{ 38 } = 5$, $\mathrm{b}_{ 40 } = 4$, $\mathrm{b}_{ 42 } = 3$, $\mathrm{b}_{ 44 } = 2$, $\mathrm{b}_{ 46 } = 1$, $\mathrm{b}_{ 48 } = 1$
$\mathrm{Aut}^0(\SGr(3,12))$
$\mathrm{PSp}_{ 12 }$
$\pi_0\mathrm{Aut}(\SGr(3,12))$
$1$
$\dim\mathrm{Aut}^0(\SGr(3,12))$
78
Projective geometry
minimal embedding

$\SGr(3,12)\hookrightarrow\mathbb{P}^{ 207 }$

degree
135980416
Hilbert series
1, 208, 13650, 448800, 9145422, 130546416, 1412391981, 12228502000, 88124965500, 544437269376, 2949919035856, 14272287324672, 62557462858800, 251362113206400, 934992250975068, 3246146638562256, 10592511867819805, 32679889043230000, 95815583771943750, 268160850289332000, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
102720
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 $\mathbf D^{\mathrm b}(\SGr(3,12))$:

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 $\mathrm{c}_1(\SGr(3,12))$:

Homological projective duality

Full minimal rectangular Lefschetz collection. [Novikov, Theorem 1 and Remark 2]

Lefschetz block sizes: 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Block sizes $(16, 16, 16, 16, 16, 16, 16, 16, 16, 16)$; 160 objects.

Equivariant bundles built from Schur functors of the tautological bundle, symplectic exterior powers, and staircase complexes.

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. Novikov, A. A. Minimal Lefschetz collections on isotropic Grassmannians $IGr(3,2n)$. Russian Math. Surveys 80 (2025), 543–545. MR4961293.
  5. Lyalya Guseva; Alexander Novikov. Full exceptional collections on the symplectic isotropic Grassmannians. Advances in Mathematics (2026), article 111211. arXiv:2505.13005.
  6. 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.