Rhombic tilings of polygons and classes of reduced words in coxeter groups

[摘要] In the standard Coxeter presentation, the symmetric group S-n is generated by the adjacent 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, Lye 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 diagrams to prove an enumerative conjecture by Kuperberg and Propp, as well as a ii-analogue thereof Finally, we give similar constructions for two other families of finite Coxeter groups, namely those of types B and D. (C) 1997 Academic Press.

[发布日期] 1997-02-01