McKay Arrow Graph,Loewy Matrix And Dimension Vector

This paper studies the dimension vectors complexity of the Loewy matrix for two-dimensional Mckay quiver.At first,we mainly obtain the ?-matrix equivalent diagonal matrix D(l) and its related elementary transformation matrix E(l) by applying the Euclid ring matrix diagonal method.Then,the base vector of Jordan block in the Loewy matrix of eigenvalue 1 can be calculated.We further calculate the complexity of these base vectors and these linear combinations.We can get the characterization of positive vectors of different complexity.Finally we explore the L-invariant vectors and the C-invariant vectors for A4,D4,E6,E7,E8 Mckay matrix and give the relationship between their invariant vectors.