\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.

The Freudenthal magic square

varieties projective geometric description
$\nu_4(\mathrm{Q}^1)\hookrightarrow\mathbb{P}^{ 4 }$
\begin{align*} d&=0 \\ i&=0 \\ \chi&=0 \end{align*}
$\mathrm{Fl}(1,2;3)\hookrightarrow\mathbb{P}^{ 7 }$
\begin{align*} d&=0 \\ i&=0 \\ \chi&=0 \end{align*}
$\SGr(2,6)\hookrightarrow\mathbb{P}^{ 13 }$
\begin{align*} d&=7 \\ i&=5 \\ \chi&=12 \end{align*}
$\mathrm{F}_{4}/\mathrm{P}_{4}\hookrightarrow\mathbb{P}^{ 25 }$
\begin{align*} d&=15 \\ i&=11 \\ \chi&=24 \end{align*}
hyperplane section of Severi varieties
$\nu_2(\mathbb{P}^2)\hookrightarrow\mathbb{P}^{ 5 }$
\begin{align*} d&=0 \\ i&=0 \\ \chi&=0 \end{align*}
$\mathbb{P}^2\times\mathbb{P}^2\hookrightarrow\mathbb{P}^{ 8 }$
\begin{align*} d&=0 \\ i&=0 \\ \chi&=0 \end{align*}
$\Gr(2,6)\hookrightarrow\mathbb{P}^{ 14 }$
\begin{align*} d&=8 \\ i&=6 \\ \chi&=15 \end{align*}
$\mathbb{OP}^2\hookrightarrow\mathbb{P}^{ 26 }$
\begin{align*} d&=16 \\ i&=12 \\ \chi&=27 \end{align*}
Severi varieties
$\LGr(3,6)\hookrightarrow\mathbb{P}^{ 13 }$
\begin{align*} d&=6 \\ i&=4 \\ \chi&=8 \end{align*}
$\Gr(3,6)\hookrightarrow\mathbb{P}^{ 19 }$
\begin{align*} d&=9 \\ i&=6 \\ \chi&=20 \end{align*}
$\OGr_+(6,12)\hookrightarrow\mathbb{P}^{ 31 }$
\begin{align*} d&=15 \\ i&=10 \\ \chi&=32 \end{align*}
$\mathrm{E}_{7}/\mathrm{P}_{7}\hookrightarrow\mathbb{P}^{ 55 }$
\begin{align*} d&=27 \\ i&=18 \\ \chi&=56 \end{align*}
lines through a point of the exceptional adjoints
$\mathrm{F}_{4}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 51 }$
\begin{align*} d&=15 \\ i&=8 \\ \chi&=24 \end{align*}
$\mathrm{E}_{6}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 77 }$
\begin{align*} d&=21 \\ i&=11 \\ \chi&=72 \end{align*}
$\mathrm{E}_{7}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 132 }$
\begin{align*} d&=33 \\ i&=17 \\ \chi&=126 \end{align*}
$\mathrm{E}_{8}/\mathrm{P}_{8}\hookrightarrow\mathbb{P}^{ 247 }$
\begin{align*} d&=57 \\ i&=29 \\ \chi&=240 \end{align*}
exceptional adjoints
marked Dynkin diagrams
Lie algebras
𝖘𝖔3 𝖘𝖑3 𝖘𝖕6 𝖋4
𝖘𝖑3 𝖘𝖑3x𝖘𝖑3 𝖘𝖑6 𝖊6
𝖘𝖕6 𝖘𝖑6 𝖘𝖔12 𝖊7
𝖋4 𝖊6 𝖊7 𝖊8

Freudenthal's $3\times 3$ square

There also exists a $3\times 3$ version of Freudenthal's magic square for all $n\geq 3$, where the $n=3$ corresponds to the upper left subsquare of Freudenthal's magic square.

varieties projective geometric description
$\mathrm{Q}^{1}\hookrightarrow\mathbb{P}^{ 4 }$
\begin{align*} d&=1 \\ i&=2 \\ \chi&=2 \end{align*}
$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{2}})\hookrightarrow\mathbb{P}^{ 7 }$
\begin{align*} d&=3 \\ i&=2 \\ \chi&=6 \end{align*}
$\SGr(2,6)\hookrightarrow\mathbb{P}^{ 13 }$
\begin{align*} d&=7 \\ i&=5 \\ \chi&=12 \end{align*}
hyperplane section
$\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{ 5 }$
\begin{align*} d&=2 \\ i&=2 \\ \chi&=3 \end{align*}
$\mathbb{P}^{2}\times\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{ 8 }$
\begin{align*} d&=4 \\ i&=3 \\ \chi&=9 \end{align*}
$\Gr(2,6)\hookrightarrow\mathbb{P}^{ 14 }$
\begin{align*} d&=8 \\ i&=6 \\ \chi&=15 \end{align*}
$\LGr(3,6)\hookrightarrow\mathbb{P}^{ 13 }$
\begin{align*} d&=6 \\ i&=4 \\ \chi&=8 \end{align*}
$\Gr(3,6)\hookrightarrow\mathbb{P}^{ 19 }$
\begin{align*} d&=9 \\ i&=6 \\ \chi&=20 \end{align*}
$\OGr_+(6,12)\hookrightarrow\mathbb{P}^{ 31 }$
\begin{align*} d&=15 \\ i&=10 \\ \chi&=32 \end{align*}
marked Dynkin diagrams
Lie algebras
𝖘𝖔3 𝖘𝖑3 𝖘𝖕6
𝖘𝖑3 𝖘𝖑3x𝖘𝖑3 𝖘𝖑6
𝖘𝖕6 𝖘𝖑6 𝖘𝖔12
varieties projective geometric description
$\mathrm{Q}^{2}\hookrightarrow\mathbb{P}^{ 8 }$
\begin{align*} d&=2 \\ i&=2 \\ \chi&=4 \end{align*}
$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{3}})\hookrightarrow\mathbb{P}^{ 14 }$
\begin{align*} d&=5 \\ i&=3 \\ \chi&=12 \end{align*}
$\SGr(2,8)\hookrightarrow\mathbb{P}^{ 26 }$
\begin{align*} d&=11 \\ i&=7 \\ \chi&=24 \end{align*}
hyperplane section
$\mathbb{P}^{3}\hookrightarrow\mathbb{P}^{ 9 }$
\begin{align*} d&=3 \\ i&=3 \\ \chi&=4 \end{align*}
$\mathbb{P}^{3}\times\mathbb{P}^{3}\hookrightarrow\mathbb{P}^{ 15 }$
\begin{align*} d&=9 \\ i&=4 \\ \chi&=16 \end{align*}
$\Gr(2,8)\hookrightarrow\mathbb{P}^{ 27 }$
\begin{align*} d&=12 \\ i&=8 \\ \chi&=28 \end{align*}
$\LGr(4,8)\hookrightarrow\mathbb{P}^{ 41 }$
\begin{align*} d&=10 \\ i&=5 \\ \chi&=16 \end{align*}
$\Gr(4,8)\hookrightarrow\mathbb{P}^{ 69 }$
\begin{align*} d&=16 \\ i&=8 \\ \chi&=70 \end{align*}
$\OGr_+(8,16)\hookrightarrow\mathbb{P}^{ 127 }$
\begin{align*} d&=28 \\ i&=14 \\ \chi&=128 \end{align*}
marked Dynkin diagrams
Lie algebras
𝖘𝖔4 𝖘𝖑4 𝖘𝖕8
𝖘𝖑4 𝖘𝖑4x𝖘𝖑4 𝖘𝖑8
𝖘𝖕8 𝖘𝖑8 𝖘𝖔16
varieties projective geometric description
$\mathrm{Q}^{3}\hookrightarrow\mathbb{P}^{ 13 }$
\begin{align*} d&=3 \\ i&=3 \\ \chi&=4 \end{align*}
$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{4}})\hookrightarrow\mathbb{P}^{ 23 }$
\begin{align*} d&=7 \\ i&=4 \\ \chi&=20 \end{align*}
$\SGr(2,10)\hookrightarrow\mathbb{P}^{ 43 }$
\begin{align*} d&=15 \\ i&=9 \\ \chi&=40 \end{align*}
hyperplane section
$\mathbb{P}^{4}\hookrightarrow\mathbb{P}^{ 14 }$
\begin{align*} d&=4 \\ i&=4 \\ \chi&=5 \end{align*}
$\mathbb{P}^{4}\times\mathbb{P}^{4}\hookrightarrow\mathbb{P}^{ 24 }$
\begin{align*} d&=16 \\ i&=5 \\ \chi&=25 \end{align*}
$\Gr(2,10)\hookrightarrow\mathbb{P}^{ 44 }$
\begin{align*} d&=16 \\ i&=10 \\ \chi&=45 \end{align*}
$\LGr(5,10)\hookrightarrow\mathbb{P}^{ 131 }$
\begin{align*} d&=15 \\ i&=6 \\ \chi&=32 \end{align*}
$\Gr(5,10)\hookrightarrow\mathbb{P}^{ 251 }$
\begin{align*} d&=25 \\ i&=10 \\ \chi&=252 \end{align*}
$\OGr_+(10,20)\hookrightarrow\mathbb{P}^{ 511 }$
\begin{align*} d&=45 \\ i&=18 \\ \chi&=512 \end{align*}
marked Dynkin diagrams
Lie algebras
𝖘𝖔5 𝖘𝖑5 𝖘𝖕10
𝖘𝖑5 𝖘𝖑5x𝖘𝖑5 𝖘𝖑10
𝖘𝖕10 𝖘𝖑10 𝖘𝖔20
varieties projective geometric description
$\mathrm{Q}^{4}\hookrightarrow\mathbb{P}^{ 19 }$
\begin{align*} d&=4 \\ i&=4 \\ \chi&=6 \end{align*}
$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{5}})\hookrightarrow\mathbb{P}^{ 34 }$
\begin{align*} d&=9 \\ i&=5 \\ \chi&=30 \end{align*}
$\SGr(2,12)\hookrightarrow\mathbb{P}^{ 64 }$
\begin{align*} d&=19 \\ i&=11 \\ \chi&=60 \end{align*}
hyperplane section
$\mathbb{P}^{5}\hookrightarrow\mathbb{P}^{ 20 }$
\begin{align*} d&=5 \\ i&=5 \\ \chi&=6 \end{align*}
$\mathbb{P}^{5}\times\mathbb{P}^{5}\hookrightarrow\mathbb{P}^{ 35 }$
\begin{align*} d&=25 \\ i&=6 \\ \chi&=36 \end{align*}
$\Gr(2,12)\hookrightarrow\mathbb{P}^{ 65 }$
\begin{align*} d&=20 \\ i&=12 \\ \chi&=66 \end{align*}
$\LGr(6,12)\hookrightarrow\mathbb{P}^{ 428 }$
\begin{align*} d&=21 \\ i&=7 \\ \chi&=64 \end{align*}
$\Gr(6,12)\hookrightarrow\mathbb{P}^{ 923 }$
\begin{align*} d&=36 \\ i&=12 \\ \chi&=924 \end{align*}
$\OGr_+(12,24)\hookrightarrow\mathbb{P}^{ 2047 }$
\begin{align*} d&=66 \\ i&=22 \\ \chi&=2048 \end{align*}
marked Dynkin diagrams
Lie algebras
𝖘𝖔6 𝖘𝖑6 𝖘𝖕12
𝖘𝖑6 𝖘𝖑6x𝖘𝖑6 𝖘𝖑12
𝖘𝖕12 𝖘𝖑12 𝖘𝖔24
varieties projective geometric description
$\mathrm{Q}^{5}\hookrightarrow\mathbb{P}^{ 26 }$
\begin{align*} d&=5 \\ i&=5 \\ \chi&=6 \end{align*}
$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{6}})\hookrightarrow\mathbb{P}^{ 47 }$
\begin{align*} d&=11 \\ i&=6 \\ \chi&=42 \end{align*}
$\SGr(2,14)\hookrightarrow\mathbb{P}^{ 89 }$
\begin{align*} d&=23 \\ i&=13 \\ \chi&=84 \end{align*}
hyperplane section
$\mathbb{P}^{6}\hookrightarrow\mathbb{P}^{ 27 }$
\begin{align*} d&=6 \\ i&=6 \\ \chi&=7 \end{align*}
$\mathbb{P}^{6}\times\mathbb{P}^{6}\hookrightarrow\mathbb{P}^{ 48 }$
\begin{align*} d&=36 \\ i&=7 \\ \chi&=49 \end{align*}
$\Gr(2,14)\hookrightarrow\mathbb{P}^{ 90 }$
\begin{align*} d&=24 \\ i&=14 \\ \chi&=91 \end{align*}
$\LGr(7,14)\hookrightarrow\mathbb{P}^{ 1429 }$
\begin{align*} d&=28 \\ i&=8 \\ \chi&=128 \end{align*}
$\Gr(7,14)\hookrightarrow\mathbb{P}^{ 3431 }$
\begin{align*} d&=49 \\ i&=14 \\ \chi&=3432 \end{align*}
$\OGr_+(14,28)\hookrightarrow\mathbb{P}^{ 8191 }$
\begin{align*} d&=91 \\ i&=26 \\ \chi&=8192 \end{align*}
marked Dynkin diagrams
Lie algebras
𝖘𝖔7 𝖘𝖑7 𝖘𝖕14
𝖘𝖑7 𝖘𝖑7x𝖘𝖑7 𝖘𝖑14
𝖘𝖕14 𝖘𝖑14 𝖘𝖔28