Integer quadratic forms and extensions of subsets of linearly independent roots

Abstract

We consider subsets of linearly independent roots in a certain root system \(\varPhi\). Let \(S'\) be such a subset, and let \(S'\) be associated with any Carter diagram \(\Gamma'\). The main question of the paper: what root \(\gamma \in \varPhi\) can be added to \(S'\) so that \(S' \cup \gamma\) is also a subset of linearly independent roots? This extra root \(\gamma\) is called the linkage root. The vector \(\gamma^{\nabla}\) of inner products \(\{(\gamma,\tau'_i)\mid \tau'_i \in S'\}\) is called the linkage label vector. Let \(B_{\Gamma'}\) be the Cartan matrix associated with \(\Gamma'\). It is shown that \(\gamma\) is a linkage root if and only if \(\mathscr{B}^{\vee}_{\Gamma'}(\gamma^{\nabla}) < 2\), where \(\mathscr{B}^{\vee}_{\Gamma'}\) is a quadratic form with the matrix inverse to \(B_{\Gamma'}\). The set of all linkage roots for \(\Gamma'\) is called a linkage system and is denoted by \(\mathscr{L}(\Gamma')\). The sizes of \(\mathscr{L}(\Gamma')\) and \(\mathscr{L}(\Gamma)\) are the same for diagrams \(\Gamma\) and \(\Gamma'\) that have the same rank and \(ADE\) type. Let \(W^{\vee}\) be the Weyl group of the quadratic form \(\mathscr{B}^{\vee}_{\Gamma'}\). The sizes and structure of orbits for linkage systems \(\mathscr{L}(D_l)\) and \(\mathscr{L}(D_l(a_k))\) are presented.