On finding the fundamental dominant weights of a root system
Let \(\Phi\) be a root system of rank \(\ell\) with fixed basis \((\alpha_1,\ldots ,\alpha_{\ell})\) and dual basis \((\lambda_1,\ldots ,\lambda_{\ell})\). Denote by \(\Phi^+=\{\alpha \in \Phi \mid \alpha=\sum_{j=1}^{\ell}k_j(\alpha)\alpha_j, k_j(\alpha)\in \mathbb N_0 \}\) the set of positive roots. As usual let \(\delta ={1\over 2}\sum_{\alpha \in \Phi^+}\alpha\). Now fix any \(j \in \{1,\ldots ,\ell\}\) and let \(\Phi^+_0=\{\alpha \in \Phi^+ \mid k_j(\alpha)=0\}\), \(\delta_0 ={1\over 2}\sum_{\alpha \in \Phi^+_0}\alpha\) and \(\delta_1=\delta-\delta_0\). Then \(\Phi_0=\Phi^+_0 \cup (-\Phi^+_0)\) forms a root system of rank \(\ell -1\). The author proves that \(\lambda_j={\delta_1 \over <\delta_1,\alpha_j>}\). This formula can be used to find the fundamental dominant weight \(\lambda_j\) for any given \(j\) without employing the Cartan matrix. The weight \(\delta_1\) is determined for all root systems of type \(A-G\).
scientific article; zbMATH DE number 1267320