English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In 1972, R. Carter introduced admissible diagrams to classify conjugacyclasses in a finite Weyl group W. We say that an admissible diagram \Gamma is aCarter diagram if any edge {\alpha, \beta} with inner product (\alpha, \beta) >0 (resp. (\alpha, \beta) < 0) is drawn as dotted (resp. solid) edge. Weconstruct an explicit transformation of any Carter diagram containing longcycles (with the number of vertices l > 4) into another Carter diagramcontaining only 4-cycles. Thus, all Carter diagrams containing long cycles canbe eliminated from the classification list. There exist diagrams determiningtwo conjugacy classes in W.It is shown that any connected Carter diagram \Gammacontaining a 4-vertex pattern D_4 or D_4(a_1) determines a single conjugacyclass. 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 \leq3. Let a socket be the set of vertices of \Gamma connected to \alpha. If thenumber of sockets available for extensions is equal to 2, then there is a pairof extensions \Gamma < \tilde{\Gamma}_L and \Gamma < \tilde{\Gamma}_R, calledmirror 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 someT \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 oftwo involutions. One of the goals of this paper and its sequels is to preparethe notions and framework in which we give the proof of this fact withoutappealing to the classification of conjugacy classes.

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