Classification of linkage systems

Abstract

A linkage diagram is obtained from the Carter diagram $\Gamma$ by adding an extra root $\gamma$, so that the resulting subset of roots is linearly independent. With every linkage diagram we associate the linkage label vector ${\gamma }^{\nabla }$, similar to Dynkin labels. The linkage diagrams connected under the action of the group $W_S^{\vee }$ constitute the the linkage system $\mathcal{L}(\Gamma )$. For any simply-laced Carter diagram, the system $\mathcal{L}(\Gamma )$ is constructed. To obtain linkage diagrams ${\theta }^{\nabla }$, we use an easily verifiable criterion: ${\mathcal{B}}_{\Gamma }^{\vee }({\theta }^{\nabla })<2$, where ${\mathcal{B}}_{\Gamma }^{\vee }$ is the inverse quadratic form associated with $\Gamma$. A Dynkin diagram ${\Gamma }^{\prime }$ such that rank(${\Gamma }^{\prime }$) = rank($\Gamma$) + 1 and any $\Gamma$-associated root subset $S$ lies in $\Phi ({\Gamma }^{\prime })$, is said to be the Dynkin extension. The linkage system $\mathcal{L}(\Gamma )$ is the union of ${\Gamma }_i$-components ${\mathcal{L}}_{{\Gamma }_i}(\Gamma )$ taken for all Dynkin extensions of $\Gamma {<}_D{\Gamma }_i$. The subset $\Phi (S)$ of roots of $\Phi ({\Gamma }^{\prime })$, linearly dependent on roots of $S$ is said to be a partial root system. The size of ${\mathcal{L}}_{{\Gamma }^{\prime }}(\Gamma )$ is estimated as follows: $|{\mathcal{L}}_{{\Gamma }^{\prime }}(\Gamma )|\le |\Phi ({\Gamma }^{\prime })|-|\Phi (S)|$. 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 {$\Gamma ,\tilde{\Gamma }$} of covalent Carter diagrams, where $\Gamma$ is the Dynkin diagram, we explicitly construct the invertible linear map $M:\mathcal{P}⟶\mathcal{R}$, where $\mathcal{R}$ (resp. $\mathcal{P}$) is the root system (resp. partial root system) corresponding to $\Gamma$ (resp. $\tilde{\Gamma }$). In particular, we have $|\mathcal{L}(\tilde{\Gamma })|=|\mathcal{L}(\Gamma )|$.

118 pages, 66 figures. Updated abstract, added index. arXiv admin note: text overlap with arXiv:1010.5684

References in corpus (1)

• Root systems and diagram calculus. III. Semi-Coxeter orbits of linkage diagrams and the Carter theorem