Root Systems of Reflection Systems, and W-Graphs over Non-Commutative Algebras

We study two objects commonly associated to Coxeter systems: root systems and Hecke algebras. First, we show that there is a collection of groups (which include Coxeter groups) for which we can associate a more generalized notion of root systems. We conjecture that these groups admit a presentation similar to that of Coxeter groups, but in which generators might have order different than two. More precisely, we conjecture that these groups (together with their generating sets) are reflection systems.

Secondly, we generalize the notion of W-graphs, and use the path algebras associated to these graphs to construct representations of Hecke algebras on quotients of these path algebras. We provide an algorithm to describe the generators of the ideals we use to construct the quotients. Finally, we show how this construction is a special example of a more general framework: D-graphs over non-commutative algebras.