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Classification of linkage systems

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 $γ^{\nabla}$, similar to Dynkin labels. The linkage diagrams connected under the action of the group $W^{\vee}_{S}$ constitute the the linkage system $\mathscr{L}(Γ)$. For any simply-laced Carter diagram, the system $\mathscr{L}(Γ)$ is constructed. To obtain linkage diagrams $θ^{\nabla}$, we use an easily verifiable criterion: $\mathscr{B}^{\vee}_Γ(θ^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_Γ$ 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 $\mathscr{L}(Γ)$ is the union of $Γ_i$-components $\mathscr{L}_{Γ_i}(Γ)$ taken for all Dynkin extensions of $Γ<_D Γ_i$. 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 $\mathscr{L}_{Γ'}(Γ)$ is estimated as follows: $|\mathscr{L}_{Γ'}(Γ)| \leq |\varPhi(Γ')| - |\varPhi(S)|$. Carter diagrams $E_l$ and $E_l(a_i)$ (resp. $D_l$ and $D_l(a_k)$) 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 $M : \mathcal{P} \longrightarrow \mathcal{R}$, where $\mathcal{R}$ (resp. $\mathcal{P}$) is the root system (resp. partial root system) corresponding to $Γ$ (resp. $\widetildeΓ$). In particular, we have $|\mathscr{L}(\widetildeΓ)| = |\mathscr{L}(Γ)|$.