Grassmannian.info

A periodic table of (generalised) Grassmannians.

The Freudenthal magic square

varieties				projective geometric description
$\nu_2(\mathrm{Q}^1)\hookrightarrow\mathbb{P}^{ 4 }$ \begin{equation} \begin{aligned} d&=1 \\ i&=2 \\ \chi&=2 \end{aligned} \end{equation}	$\mathrm{Fl}(1,2;3)\hookrightarrow\mathbb{P}^{ 7 }$ \begin{equation} \begin{aligned} d&=3 \\ i&=2 \\ \chi&=6 \end{aligned} \end{equation}	$\SGr(2,6)\hookrightarrow\mathbb{P}^{ 13 }$ \begin{equation} \begin{aligned} d&=7 \\ i&=5 \\ \chi&=12 \end{aligned} \end{equation}	$\mathrm{F}_{4}/\mathrm{P}_{4}\hookrightarrow\mathbb{P}^{ 25 }$ \begin{equation} \begin{aligned} d&=15 \\ i&=11 \\ \chi&=24 \end{aligned} \end{equation}	hyperplane section of Severi varieties
$\nu_2(\mathbb{P}^2)\hookrightarrow\mathbb{P}^{ 5 }$ \begin{equation} \begin{aligned} d&=2 \\ i&=3 \\ \chi&=3 \end{aligned} \end{equation}	$\mathbb{P}^2\times\mathbb{P}^2\hookrightarrow\mathbb{P}^{ 8 }$ \begin{equation} \begin{aligned} d&=4 \\ i&=3 \\ \chi&=9 \end{aligned} \end{equation}	$\Gr(2,6)\hookrightarrow\mathbb{P}^{ 14 }$ \begin{equation} \begin{aligned} d&=8 \\ i&=6 \\ \chi&=15 \end{aligned} \end{equation}	$\mathbb{OP}^2\hookrightarrow\mathbb{P}^{ 26 }$ \begin{equation} \begin{aligned} d&=16 \\ i&=12 \\ \chi&=27 \end{aligned} \end{equation}	Severi varieties
$\LGr(3,6)\hookrightarrow\mathbb{P}^{ 13 }$ \begin{equation} \begin{aligned} d&=6 \\ i&=4 \\ \chi&=8 \end{aligned} \end{equation}	$\Gr(3,6)\hookrightarrow\mathbb{P}^{ 19 }$ \begin{equation} \begin{aligned} d&=9 \\ i&=6 \\ \chi&=20 \end{aligned} \end{equation}	$\OGr_+(6,12)\hookrightarrow\mathbb{P}^{ 31 }$ \begin{equation} \begin{aligned} d&=15 \\ i&=10 \\ \chi&=32 \end{aligned} \end{equation}	$\mathrm{E}_{7}/\mathrm{P}_{7}\hookrightarrow\mathbb{P}^{ 55 }$ \begin{equation} \begin{aligned} d&=27 \\ i&=18 \\ \chi&=56 \end{aligned} \end{equation}	lines through a point of the exceptional adjoints
$\mathrm{F}_{4}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 51 }$ \begin{equation} \begin{aligned} d&=15 \\ i&=8 \\ \chi&=24 \end{aligned} \end{equation}	$\mathrm{E}_{6}/\mathrm{P}_{2}\hookrightarrow\mathbb{P}^{ 77 }$ \begin{equation} \begin{aligned} d&=21 \\ i&=11 \\ \chi&=72 \end{aligned} \end{equation}	$\mathrm{E}_{7}/\mathrm{P}_{1}\hookrightarrow\mathbb{P}^{ 132 }$ \begin{equation} \begin{aligned} d&=33 \\ i&=17 \\ \chi&=126 \end{aligned} \end{equation}	$\mathrm{E}_{8}/\mathrm{P}_{8}\hookrightarrow\mathbb{P}^{ 247 }$ \begin{equation} \begin{aligned} d&=57 \\ i&=29 \\ \chi&=240 \end{aligned} \end{equation}	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{equation} \begin{aligned} d&=1 \\ i&=2 \\ \chi&=2 \end{aligned} \end{equation}	$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{2}})\hookrightarrow\mathbb{P}^{ 7 }$ \begin{equation} \begin{aligned} d&=3 \\ i&=2 \\ \chi&=6 \end{aligned} \end{equation}	$\SGr(2,6)\hookrightarrow\mathbb{P}^{ 13 }$ \begin{equation} \begin{aligned} d&=7 \\ i&=5 \\ \chi&=12 \end{aligned} \end{equation}	hyperplane section
$\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{ 5 }$ \begin{equation} \begin{aligned} d&=2 \\ i&=3 \\ \chi&=3 \end{aligned} \end{equation}	$\mathbb{P}^{2}\times\mathbb{P}^{2}\hookrightarrow\mathbb{P}^{ 8 }$ \begin{equation} \begin{aligned} d&=4 \\ i&=3 \\ \chi&=9 \end{aligned} \end{equation}	$\Gr(2,6)\hookrightarrow\mathbb{P}^{ 14 }$ \begin{equation} \begin{aligned} d&=8 \\ i&=6 \\ \chi&=15 \end{aligned} \end{equation}
$\LGr(3,6)\hookrightarrow\mathbb{P}^{ 13 }$ \begin{equation} \begin{aligned} d&=6 \\ i&=4 \\ \chi&=8 \end{aligned} \end{equation}	$\Gr(3,6)\hookrightarrow\mathbb{P}^{ 19 }$ \begin{equation} \begin{aligned} d&=9 \\ i&=6 \\ \chi&=20 \end{aligned} \end{equation}	$\OGr_+(6,12)\hookrightarrow\mathbb{P}^{ 31 }$ \begin{equation} \begin{aligned} d&=15 \\ i&=10 \\ \chi&=32 \end{aligned} \end{equation}
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{equation} \begin{aligned} d&=2 \\ i&=2 \\ \chi&=4 \end{aligned} \end{equation}	$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{3}})\hookrightarrow\mathbb{P}^{ 14 }$ \begin{equation} \begin{aligned} d&=5 \\ i&=3 \\ \chi&=12 \end{aligned} \end{equation}	$\SGr(2,8)\hookrightarrow\mathbb{P}^{ 26 }$ \begin{equation} \begin{aligned} d&=11 \\ i&=7 \\ \chi&=24 \end{aligned} \end{equation}	hyperplane section
$\mathbb{P}^{3}\hookrightarrow\mathbb{P}^{ 9 }$ \begin{equation} \begin{aligned} d&=3 \\ i&=4 \\ \chi&=4 \end{aligned} \end{equation}	$\mathbb{P}^{3}\times\mathbb{P}^{3}\hookrightarrow\mathbb{P}^{ 15 }$ \begin{equation} \begin{aligned} d&=6 \\ i&=4 \\ \chi&=16 \end{aligned} \end{equation}	$\Gr(2,8)\hookrightarrow\mathbb{P}^{ 27 }$ \begin{equation} \begin{aligned} d&=12 \\ i&=8 \\ \chi&=28 \end{aligned} \end{equation}
$\LGr(4,8)\hookrightarrow\mathbb{P}^{ 41 }$ \begin{equation} \begin{aligned} d&=10 \\ i&=5 \\ \chi&=16 \end{aligned} \end{equation}	$\Gr(4,8)\hookrightarrow\mathbb{P}^{ 69 }$ \begin{equation} \begin{aligned} d&=16 \\ i&=8 \\ \chi&=70 \end{aligned} \end{equation}	$\OGr_+(8,16)\hookrightarrow\mathbb{P}^{ 127 }$ \begin{equation} \begin{aligned} d&=28 \\ i&=14 \\ \chi&=128 \end{aligned} \end{equation}
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{equation} \begin{aligned} d&=3 \\ i&=3 \\ \chi&=4 \end{aligned} \end{equation}	$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{4}})\hookrightarrow\mathbb{P}^{ 23 }$ \begin{equation} \begin{aligned} d&=7 \\ i&=4 \\ \chi&=20 \end{aligned} \end{equation}	$\SGr(2,10)\hookrightarrow\mathbb{P}^{ 43 }$ \begin{equation} \begin{aligned} d&=15 \\ i&=9 \\ \chi&=40 \end{aligned} \end{equation}	hyperplane section
$\mathbb{P}^{4}\hookrightarrow\mathbb{P}^{ 14 }$ \begin{equation} \begin{aligned} d&=4 \\ i&=5 \\ \chi&=5 \end{aligned} \end{equation}	$\mathbb{P}^{4}\times\mathbb{P}^{4}\hookrightarrow\mathbb{P}^{ 24 }$ \begin{equation} \begin{aligned} d&=8 \\ i&=5 \\ \chi&=25 \end{aligned} \end{equation}	$\Gr(2,10)\hookrightarrow\mathbb{P}^{ 44 }$ \begin{equation} \begin{aligned} d&=16 \\ i&=10 \\ \chi&=45 \end{aligned} \end{equation}
$\LGr(5,10)\hookrightarrow\mathbb{P}^{ 131 }$ \begin{equation} \begin{aligned} d&=15 \\ i&=6 \\ \chi&=32 \end{aligned} \end{equation}	$\Gr(5,10)\hookrightarrow\mathbb{P}^{ 251 }$ \begin{equation} \begin{aligned} d&=25 \\ i&=10 \\ \chi&=252 \end{aligned} \end{equation}	$\OGr_+(10,20)\hookrightarrow\mathbb{P}^{ 511 }$ \begin{equation} \begin{aligned} d&=45 \\ i&=18 \\ \chi&=512 \end{aligned} \end{equation}
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{equation} \begin{aligned} d&=4 \\ i&=4 \\ \chi&=6 \end{aligned} \end{equation}	$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{5}})\hookrightarrow\mathbb{P}^{ 34 }$ \begin{equation} \begin{aligned} d&=9 \\ i&=5 \\ \chi&=30 \end{aligned} \end{equation}	$\SGr(2,12)\hookrightarrow\mathbb{P}^{ 64 }$ \begin{equation} \begin{aligned} d&=19 \\ i&=11 \\ \chi&=60 \end{aligned} \end{equation}	hyperplane section
$\mathbb{P}^{5}\hookrightarrow\mathbb{P}^{ 20 }$ \begin{equation} \begin{aligned} d&=5 \\ i&=6 \\ \chi&=6 \end{aligned} \end{equation}	$\mathbb{P}^{5}\times\mathbb{P}^{5}\hookrightarrow\mathbb{P}^{ 35 }$ \begin{equation} \begin{aligned} d&=10 \\ i&=6 \\ \chi&=36 \end{aligned} \end{equation}	$\Gr(2,12)\hookrightarrow\mathbb{P}^{ 65 }$ \begin{equation} \begin{aligned} d&=20 \\ i&=12 \\ \chi&=66 \end{aligned} \end{equation}
$\LGr(6,12)\hookrightarrow\mathbb{P}^{ 428 }$ \begin{equation} \begin{aligned} d&=21 \\ i&=7 \\ \chi&=64 \end{aligned} \end{equation}	$\Gr(6,12)\hookrightarrow\mathbb{P}^{ 923 }$ \begin{equation} \begin{aligned} d&=36 \\ i&=12 \\ \chi&=924 \end{aligned} \end{equation}	$\OGr_+(12,24)\hookrightarrow\mathbb{P}^{ 2047 }$ \begin{equation} \begin{aligned} d&=66 \\ i&=22 \\ \chi&=2048 \end{aligned} \end{equation}
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{equation} \begin{aligned} d&=5 \\ i&=5 \\ \chi&=6 \end{aligned} \end{equation}	$\mathbb{P}(\mathrm{T}_{\mathbb{P}^{6}})\hookrightarrow\mathbb{P}^{ 47 }$ \begin{equation} \begin{aligned} d&=11 \\ i&=6 \\ \chi&=42 \end{aligned} \end{equation}	$\SGr(2,14)\hookrightarrow\mathbb{P}^{ 89 }$ \begin{equation} \begin{aligned} d&=23 \\ i&=13 \\ \chi&=84 \end{aligned} \end{equation}	hyperplane section
$\mathbb{P}^{6}\hookrightarrow\mathbb{P}^{ 27 }$ \begin{equation} \begin{aligned} d&=6 \\ i&=7 \\ \chi&=7 \end{aligned} \end{equation}	$\mathbb{P}^{6}\times\mathbb{P}^{6}\hookrightarrow\mathbb{P}^{ 48 }$ \begin{equation} \begin{aligned} d&=12 \\ i&=7 \\ \chi&=49 \end{aligned} \end{equation}	$\Gr(2,14)\hookrightarrow\mathbb{P}^{ 90 }$ \begin{equation} \begin{aligned} d&=24 \\ i&=14 \\ \chi&=91 \end{aligned} \end{equation}
$\LGr(7,14)\hookrightarrow\mathbb{P}^{ 1429 }$ \begin{equation} \begin{aligned} d&=28 \\ i&=8 \\ \chi&=128 \end{aligned} \end{equation}	$\Gr(7,14)\hookrightarrow\mathbb{P}^{ 3431 }$ \begin{equation} \begin{aligned} d&=49 \\ i&=14 \\ \chi&=3432 \end{aligned} \end{equation}	$\OGr_+(14,28)\hookrightarrow\mathbb{P}^{ 8191 }$ \begin{equation} \begin{aligned} d&=91 \\ i&=26 \\ \chi&=8192 \end{aligned} \end{equation}
marked Dynkin diagrams



Lie algebras
𝖘𝖔₇	𝖘𝖑₇	𝖘𝖕₁₄
𝖘𝖑₇	𝖘𝖑₇x𝖘𝖑₇	𝖘𝖑₁₄
𝖘𝖕₁₄	𝖘𝖑₁₄	𝖘𝖔₂₈