On McKay Quiver Of Type G₇

This paper mainly discusses the dimension vector complexity of the Loewy matrix for three-dimensional McKay quiver of the type G₇.Firstly,the λmatrix equivalent diagonal matrix D（l）（We use Matlab to test the calculation results.）and its related transformation matrix E（l）were obtained by using Euclid ring matrix diagonal method.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.Under the condition of invariable in complexity to define a set of base vectors that each component for nonnegative integer to explore the complexity of the base vectors linear combinations,so as to obtain descriptions of positive vectors of different complexity.Lastly,we discuss the invariant vector（ie,vector with complexity of 1）under Loewy Matrix.