2011/05/14 by Rafael Stekolshchik, Stekolshchik, Rafael
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1105.2875
38 pages, 58 figures, 20 tables. Updated figures B.13-B.19 and tables B.6-B.12
arxiv created 2011/08/05 · arxiv updated 2011/08/08
A diagram obtained from the Carter diagram Γ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the \it linkage diagram. Given a linkage diagram, we associate the linkage labels vector, which is introduced like the vector of Dynkin labels. Similarly to the dual Weyl group, we introduce the group W\veeL associated with Γ, and we call it the dual partial Weyl group. The linkage labels vectors connected under the action of W\veeL constitute the linkage system \mathscrL(Γ), which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. The Carter theorem states that every element of a Weyl group W is expressible as the product of two involutions. We give the proof of this theorem based on the description of the linkage system \mathscrL(Γ) and semi-Coxeter orbits of linkage labels vectors for any Carter diagram Γ. The main idea of the proof is based on the fact that, with a few exceptions, in each semi-Coxeter orbit there is a special linkage diagram -- called \it unicolored, for which the decomposition into the product of two involutions is trivial.