On McKay Quiver Of Type E₇

This paper mainly discusses the vector complexity of the Loewy matrix for three-dimensional McKay quiver of the type E₇.First,the ?-matrix equiv-alent diagonal matrix D?l?and its related transformation matrix E?l?were obtained by elementary transformation of the ?-matrix of three-dimensional McKay quiver of type E₇.then,the elementary factors of the Loewy matrix are obtained,and we calculate the base vector of Jordan block in the Loewy matrix of eigenvalue 1.We get the complexity of these linear combinations of base vectors.The nature of the complexity then gets the positive of its different complexity For vector characterization,we discuss the invariant vec-tor?ie,vector with complexity of 1?under Loewy Matrix,and for A₄,D₄,E₆,E₇ McKay matrix gives the relationship between their invariant vectors and the corresponding Coxeter transform invariant vectors.