Root systems and diagram calculus. I. Regular extensions of Carter diagrams and the uniqueness of conjugacy classes

Abstract

In 1972, R. Carter introduced admissible diagrams to classify conjugacy classes in a finite Weyl group W. We say that an admissible diagram \Gamma is a Carter diagram if any edge {\alpha, \beta} with inner product (\alpha, \beta) > 0 (resp. (\alpha, \beta) < 0) is drawn as dotted (resp. solid) edge. We construct an explicit transformation of any Carter diagram containing long cycles (with the number of vertices l > 4) into another Carter diagram containing only 4-cycles. Thus, all Carter diagrams containing long cycles can be eliminated from the classification list. There exist diagrams determining two conjugacy classes in W.It is shown that any connected Carter diagram \Gamma containing a 4-vertex pattern D_4 or D_4(a_1) determines a single conjugacy class. The main approach is studying different extensions of Carter diagrams. Let \tilde{\Gamma} be the Carter diagram obtained from a certain Carter diagram \Gamma by adding a single vertex \alpha connected to \Gamma at n points, n \leq 3. Let a socket be the set of vertices of \Gamma connected to \alpha. If the number of sockets available for extensions is equal to 2, then there is a pair of extensions \Gamma < \tilde{\Gamma}_L and \Gamma < \tilde{\Gamma}_R, called mirror extensions and the pair elements w_L and w_R associated with \tilde{\Gamma}_L and \tilde{\Gamma}_R. We show that w_R = T^{-1}w_L{T} for some T \in W, where the map T is explicitly constructed for all mirror extensions. In Carter's description of the conjugacy classes in a Weyl group a key result (Carter's theorem) states that every element in a Weyl group is a product of two involutions. One of the goals of this paper and its sequels is to prepare the notions and framework in which we give the proof of this fact without appealing to the classification of conjugacy classes.