Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups

In the standard Coxeter presentation, the symmetric group Sn is generated by the ad jacent transpositions (1,2), (2,3), ... , (n1,n). For any given permutation, we consider all minimal-length factorizations thereof as a product of the generators. Any two transpositions (i,i+1) and (j,j+1) commute if the numbers i and j are not consecutive; thus, in any factorization, their order can be switched to obtain another factorization of the same permutation. Extending this to an equivalence relation, we establish a bijection between the resulting equivalence classes and rhombic tilings of a certain 2n-gon determined by the permutation. We also study the graph structure induced on the set of tilings by the other Coxeter relations. For a special case, we use lattice-path d iagrams to prove an enumerative conjecture by G. Kuperberg and J. Propp (counting rhombic tilings of certain octagons), as well as a q-analogue thereof. Finally, we give similar constructions for two other families of finite Coxeter groups, namely those of types B and D.