Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

Abstract

Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. We say that two reduced expressions are braid equivalent if they are related via a sequence of braid moves, and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are cubical by constructing an embedding of the braid graph into a hypercube graph whose image is an induced subgraph of the hypercube. For a special class of links, called Fibonacci links, we prove that this embedding is an isometry from the corresponding braid graph to a Fibonacci cube graph.