A wild quiver in linear systems theory

AbstractWe study the action α: ((T, L), (A, B)) ↦ (T-1AT,T-1BL) of the group Gln (K) × Glm(K) of linear state and input transformations on the space Un,m of control systems (A, B) ϵ Kn×n × Kn×m with n different eigenvalues, K an algebraically closed field. This action is of interest in systems theory as well as the representation theory of quivers. We construct a canonical form for α on Un,m and determine completely its stabilizer subgroups. The results are based on an analysis of the scaling action ((D,E), B) ↦ D-1 BE of the group of diagonal matrices Dn(K) × Dm(K) on Kn×m. We construct a canonical form for this action by graph-theoretic methods and determine the adherence order of the underlying stratification of Kn×m