مثال ها
گروه کوکسترA 1 × A 1الف 2ب 2H 2G 2{\displaystyle {\tilde {I}}_{1}}الف 3ب 3د 4آ~3{\displaystyle {\tilde {A}}_{3}}
نمودار کاکستر
ماتریس کوکستر{\displaystyle \left[{\begin{smallmatrix}1&2\\2&1\\\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&3\\3&1\\\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&4\\4&1\\\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&5\\5&1\\\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&6\\6&1\\\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&\infty \\\infty &1\\\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&3&2\\3&1&3\\2&3&1\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&4&2\\4&1&3\\2&3&1\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&3&2&2\\3&1&3&3\\2&3&1&2\\2&3&2&1\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}1&3&2&3\\3&1&3&2\\2&3&1&3\\3&2&3&1\end{smallmatrix}}\right]}
ماتریس شلافلی{\displaystyle \left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-1\\-1&\ \,2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-{\sqrt {2}}\\-{\sqrt {2}}&\ \,2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-\phi \\-\phi &\ \,2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-{\sqrt {3}}\\-{\sqrt {3}}&\ \,2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-2\\-2&\ \,2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-1&\ \,0\\-1&\ \,2&-1\\\ \,0&-1&\ \,2\end{smallmatrix}} \درست]}{\displaystyle \left[{\begin{smallmatrix}\ \,\ \ 2&-{\sqrt {2}}&\ \,0\\-{\sqrt {2}}&\ \,\ \ 2&-1 \\\ \،\ \ 0&\ \,-1&\ \,2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-1&\ \,0&\ \,0\\-1&\ \,2&-1&-1\\\ \,0&-1&\ \,2& \ \,0\\\ \,0&-1&\ \,0&\ \,2\end{smallmatrix}}\راست]}{\displaystyle \left[{\begin{smallmatrix}\ \,2&-1&\ \,0&-1\\-1&\ \,2&-1&\ \,0\\\ \,0&-1&\ \,2& -1\\-1&\ \,0&-1&\ \,2\end{smallmatrix}}\راست]}