Classes of maximal-length reduced words in Coxeter groups
This thesis is concerned with the graph R of all reduced words for the longest element in the Coxeter groups of classical type, the edges representing braid relations. In chapter 2 we consider the equivalence relation on the vertices of R generated by commuting adjacent letters if the corresponding simple reflections commute. An inductive way of describing all of the resulting commutation classes is described. A characterisation of the quiver-compatible commutation classes, couched in terms of letter-multiplicities, is presented in chapter 3. Chapter 4 introduces for each positive root ß an equivalence relation on R whose equivalence dasses are connected subgraphs called ß-components. It is shown that the ß-components are in bijective correspondence with the root vectors for ß (following Bedard) when ß is the highest root ∝0; in general there are more ß-components. It happens that the natural quotient graph of ß-components is determined up to isomorphism by the length of ß; we choose to focus on the ∝0-components. In chapter 5 we show that each ∝0-component in type A1, contains a unique quiver-compatible commutation class. In chapter 6 we count the ∝0-components in type B1, by exhibiting explicit representatives which have a natural interpretation as partial quivers. The edges of the graphs of ∝0-components in types A1, and B1, are determined by interpreting maximal chains in certain posets as elements of the Coxeter group of type A1-2 or B1-2, respectively. Chapter 7 establishes an isomorphism between the graphs of ß-components in types B1 and D1 whenever ß is long.