In the classification of Coxeter groups, or equivalently root systems: $$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$ with $p \geq 7$, the last four fail to generate any simple finite dimensional Lie algebras over fields of characteristic zero because of the crystallographic restriction theorem. However, I know that there exist […]

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By “reflection” I mean reflection in a hyperplane: the isometry fixing a hyperplane and moving every other point along the orthogonal line joining it to the hyperplane to the same distance on the other side. Every […]

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn…). It’s obvious that $A_k^2=1$. Now have a look a the right coloring: You’ll see that $(A_dA_r)^2=(A_rA_b)^2=(A_bA_d)^2=1$ as well. Let’s use $a,b,c$ instead. We summarize this as: […]

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter diagram is clearer, simpler, or more useful than a more explicit notation. The information on Wikipedia has not helped me. Wikipedia tells me, for […]

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