# Grassmannian.info

A periodic table of (generalised) Grassmannians.

## Horospherical varieties of rank one

• has been constructed
• has not yet been constructed
• semisimple
• not semisimple
• unknown
• generically semisimple
• unknown
 Y Z Y Z Y Z B3 X 1(3) \begin{align*} d&=9 \\ i&=5 \\ \chi&=20 \end{align*} X 1(4) \begin{align*} d&=14 \\ i&=6 \\ \chi&=48 \end{align*} X 1(5) \begin{align*} d&=20 \\ i&=7 \\ \chi&=112 \end{align*} X 1(6) \begin{align*} d&=27 \\ i&=8 \\ \chi&=256 \end{align*} X 1(7) \begin{align*} d&=35 \\ i&=9 \\ \chi&=576 \end{align*}
 Y Z Y Z Y Z B3 X 2 \begin{align*} d&=9 \\ i&=7 \\ \chi&=14 \end{align*}
 Y Y Y Y Y Y Z Z Z Z Z Z Y Y Y Y Y Y Z Z Z Z Z Z Y Y Y Y Y Z Z Z Z Z C2 Y Y Y Y Z Z Z Z C3 Y Y Y Z Z Z C4 Y Y Z Z C5 Y Z C6 C7 X 3(2, 2) \begin{align*} d&=5 \\ i&=4 \\ \chi&=8 \end{align*} X 3(3, 2) \begin{align*} d&=9 \\ i&=6 \\ \chi&=18 \end{align*} X 3(3, 3) \begin{align*} d&=9 \\ i&=5 \\ \chi&=20 \end{align*} X 3(4, 2) \begin{align*} d&=13 \\ i&=8 \\ \chi&=32 \end{align*} X 3(4, 3) \begin{align*} d&=15 \\ i&=7 \\ \chi&=56 \end{align*} X 3(4, 4) \begin{align*} d&=14 \\ i&=6 \\ \chi&=48 \end{align*} X 3(5, 2) \begin{align*} d&=17 \\ i&=10 \\ \chi&=50 \end{align*} X 3(5, 3) \begin{align*} d&=21 \\ i&=9 \\ \chi&=120 \end{align*} X 3(5, 4) \begin{align*} d&=22 \\ i&=8 \\ \chi&=160 \end{align*} X 3(5, 5) \begin{align*} d&=20 \\ i&=7 \\ \chi&=112 \end{align*} X 3(6, 2) \begin{align*} d&=21 \\ i&=12 \\ \chi&=72 \end{align*} X 3(6, 3) \begin{align*} d&=27 \\ i&=11 \\ \chi&=220 \end{align*} X 3(6, 4) \begin{align*} d&=30 \\ i&=10 \\ \chi&=400 \end{align*} X 3(6, 5) \begin{align*} d&=30 \\ i&=9 \\ \chi&=432 \end{align*} X 3(6, 6) \begin{align*} d&=27 \\ i&=8 \\ \chi&=256 \end{align*} X 3(7, 2) \begin{align*} d&=25 \\ i&=14 \\ \chi&=98 \end{align*} X 3(7, 3) \begin{align*} d&=33 \\ i&=13 \\ \chi&=364 \end{align*} X 3(7, 4) \begin{align*} d&=38 \\ i&=12 \\ \chi&=840 \end{align*} X 3(7, 5) \begin{align*} d&=40 \\ i&=11 \\ \chi&=1232 \end{align*} X 3(7, 6) \begin{align*} d&=39 \\ i&=10 \\ \chi&=1120 \end{align*} X 3(7, 7) \begin{align*} d&=35 \\ i&=9 \\ \chi&=576 \end{align*}
 Y Z Y Z Y Z Y Z F4 X 4 \begin{align*} d&=23 \\ i&=6 \\ \chi&=192 \end{align*}
 Y Z Y Z G2 X 5 \begin{align*} d&=7 \\ i&=4 \\ \chi&=12 \end{align*}