Coxeter Group

A group generated by the elements $P_i$ for $i=1$, ..., $n$ subject to

$$(P_iP_j)^{M_{ij}}=1,$$

where $M_{ij}$ are the elements of a Coxeter Matrix. Coxeter used the Notation $[3^{p,q,r}]$ for the Coxeter group generated by the nodes of a Y-shaped Coxeter-Dynkin Diagram whose three arms have $p$, $q$, and $r$ Edges. A Coxeter group of this form is finite Iff

$${1\over p+1}+{1\over q+1}+{1\over r+1}>1.$$

See also Bimonster

References

Arnold, V. I. ``Snake Calculus and Combinatorics of Bernoulli, Euler, and Springer Numbers for Coxeter Groups.'' Russian Math. Surveys 47, 3-45, 1992.