RHOMBIC TILINGS OF POLYGONS AND CLASSES OF REDUCED WORDS IN
COXETER GROUPS
Serge Elnitsky
In the standard Coxeter presentation, the symmetric group Sn 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, we study the resulting equivalence classes, and establish a
bijection between those 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 other Coxeter relations. For a special case, we
use lattice path diagrams to prove an enumerative conjecture by G.
Kuperberg and J. Propp, 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.
Doctoral Committee:
Associate Professor John R. Stembridge, Chair
Professor John E. Fornaess
Professor Robert L. Griess, Jr.
Assistant Professor William Jockusch
Professor Michael B. Woodroofe
