Involutions and reflection subgroups of finite Coxeter groups.
This paper, published after the author's death in 2004, gives a ``purely algebraic'' derivation of the classification of the finite Coxeter groups -- namely those finite groups generated by involutions \(s_i\) subject to defining relations \({s_i}^2=(s_is_j)^{m_{ij}}=1\) where the \(m_{ij}\) are specified integers. The approach, which involves a fairly detailed analysis of multiplication of words in the generators, is not surprising; other authors have indicated how this approach (or aspects of it) would work. A more standard and much easier proof of the classification uses geometric representations. Here the classification depends on Proposition 2, which asserts that the order of each product \(s_is_j\) of a pair of generators is equal to the exponent \(m_{ij}\) given in the group presentation, or is infinite when \(m_{ij}\) is unspecified. This is proved in an addendum [see Eur. J. Comb. 30, No. 8, 1913-1918 (2009; Zbl 1191.20038)].
scientific article; zbMATH DE number 5641715