Speaker
Description
We construct a general theory of finite groups generated by reflections based on the notion of numberings of partially ordered sets. The classical theory of Coxeter groups (for the case of the symmetric group) corresponds to a special choice of an ordered set --- the simplest Young diagram $(n,1)$. From a formal viewpoint, we replace the defining relation $(\sigma_i \cdot \sigma_{i + 1})^3 = Id$, where $\sigma_i, i = 1,2 ... k$, are involutions, by the relation $(\sigma_i \cdot \sigma_{i + 1})^6 = Id$ leaving the commutation condition unchanged: $(\sigma_i \cdot \sigma_j)^2 = Id$ for $|i-j|>1$, with the following extra condition: the group generated by the adjacent involutions $\sigma_i$ and $\sigma_{i+1}$ is a finite product of the groups of orders $2$ and $3$. Such symmetry groups (finite and countable) naturally arise in combinatorics and their classification does not seem hopeless. For example, if the ordered set is a finite Young diagram, then hypothetically we do not go beyond the Coxeter groups. The characteristic of our approach is that groups are considered as special subgroups of symmetric groups. Particularly interesting are infinite such groups and their representations.
