Symmetries of quivers and Coxeter transformations (Q1336468)

Let \(\Delta=(\Delta_0,\Delta_1)\) be a connected quiver (a finite oriented graph) without oriented cycles, where \(\Delta_0\) and \(\Delta_1\) is the set of vertices and the set of arrows of \(\Delta\), respectively. A symmetry of \(\Delta\) is a permutation of the set \(\Delta_0\) which induces an automorphism of the quiver \(\Delta\). The set of all symmetries of \(\Delta\) forms a group \(\text{Aut}(\Delta)\). Each symmetry \(g\in\text{Aut}(\Delta)\) gives rise to a matrix \(\widehat{g}\in\text{Gl}(|\Delta_0|,\mathbb{C})\). For any subgroup \(G\) of \(\text{Aut}(\Delta)\) the map \(g\mapsto\widehat {g}\) defines a representation \(\gamma:G\to\text{Gl}(|\Delta_0|,\mathbb{C})\), called the canonical representation of \(G\). We recall that the Coxeter matrix of \(\Delta\) is the integral \(|\Delta_0|\times|\Delta_0|\)-matrix \(\Phi_\Delta=-C^{-1}_\Delta C_\Delta\), where \(C_\Delta=[c_{ij}]\) is the Cartan matrix of \(\Delta\) and \(c_{ij}\) is the number of oriented paths from the vertex \(j\in\Delta_0\) to the vertex \(i\in\Delta_0\). Note that \(C_\Delta\) is invertible, because the trivial empty path is the only oriented path from \(i\) to \(i\), and therefore \(c_{ii}=1\) for any \(i\in\Delta_0\). The spectrum of \(\Phi_\Delta\) is the set \(\text{Spec}(\Phi_\Delta)\) of all eigenvalues of \(\Phi_\Delta\). In the reviewed paper relations between the canonical representation \(\gamma:G\to\text{Gl}(|\Delta_0|,\mathbb{C})\) and the spectrum \(\text{Spec}(\Phi_\Delta)\) of \(\Phi_\Delta\) are studied. On one side, from the indecomposable rational or complex decomposition of the representation \(\gamma\) one derives information about \(\text{Spec}(\Phi_\Delta)\). On the other side, from the Jordan form of \(\Phi_\Delta\) one derives some restrictions on the group \(\text{Aut}(\Delta)\) and decompositions of the canonical representation \(\gamma\).