2014/06/11 by Rafael Stekolshchik, Stekolshchik, Rafael
Mathematics · #20C33 #22E40 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:20C33 #msc:22E40
paper · pdf · doi:10.48550/arxiv.1406.3049
118 pages, 66 figures. Updated abstract, added index. arXiv admin note: text overlap with arXiv:1010.5684
arxiv created 2014/06/15 · arxiv updated 2014/06/17
A linkage diagram is obtained from the Carter diagram Γ by adding an extra root γ, so that the resulting subset of roots is linearly independent. With every linkage diagram we associate the linkage label vector γ∇, similar to Dynkin labels. The linkage diagrams connected under the action of the group W\veeS constitute the the linkage system \mathscrL(Γ). For any simply-laced Carter diagram, the system \mathscrL(Γ) is constructed. To obtain linkage diagrams θ∇, we use an easily verifiable criterion: \mathscrB\veeΓ(θ∇) < 2, where \mathscrB\veeΓ is the inverse quadratic form associated with Γ. A Dynkin diagram Γ' such that rank(Γ') = rank(Γ) + 1 and any Γ-associated root subset S lies in \varPhi(Γ'), is said to be the Dynkin extension. The linkage system \mathscrL(Γ) is the union of Γi-components \mathscrLΓi(Γ) taken for all Dynkin extensions of Γ<D Γi. The subset \varPhi(S) of roots of \varPhi(Γ'), linearly dependent on roots of S is said to be a partial root system. The size of \mathscrLΓ'(Γ) is estimated as follows: |\mathscrLΓ'(Γ)| ≤ |\varPhi(Γ')| - |\varPhi(S)|. Carter diagrams El and El(ai) (resp. Dl and Dl(ak)) are said to be covalent. For any pair Γ, \widetildeΓ of covalent Carter diagrams, where Γ is the Dynkin diagram, we explicitly construct the invertible linear map M : P \longrightarrow R, where R (resp. P) is the root system (resp. partial root system) corresponding to Γ (resp. \widetildeΓ). In particular, we have |\mathscrL(\widetildeΓ)| = |\mathscrL(Γ)|.