Loewy Matrix And Hierarchical Dimension Vector For Three-dimensional McKay Quiver Of Type ???₈

The study subject in this paper is three-dimensional MaKay quiver of the type E8,which we obtained by adding one circle at each vertex for the Heavy ar-row of the Affine Dynkin graphs Eg.we mainly discusses the dimension vectors complexity of the Loewy matrix for this MaKay quiver.At first,we obtain the?-matrix equivalent diagonal matrix D(l)and its related transformation ma-trix E(l)of three-dimensional Mckay quiver of type E8.We can naturally get the invariant factors and elementary factors,Then,the base vectors of Jordan block in the Loewy matrix of eigenvalue 1 can be calculated.We further cal-culate and discuss some important properties about complexity of these base vectors,Finally we study the invariant vector under Loewy Matrix called the L-invariant vectors and give the corresponding relations with the C-invariant vectors.