Study On The Properties Of Loewy Matrix Of The Three-dimensional McKay Quiver Of Type (?)

In 1980,McKay in[15]came up with the McKay quiver.Let G be a finite subgroup of GL(2,k),McKay quiver Q of G is the echo quiver of (?) of the affine Dynkin diagrams based on the underlying graphs.Since McKay quiver(McKay corresponding)is very important in mathematics,to calculate the complexity of the module of Koszul selfinjective algebra brought forward by it is important.This master degree papers mainly discuss the character of Loewy Matrix of three-dimensional McKay quiver of type.(?).Discussecd in this paper are a finite subgroup G of GL(2,k)limited in GL(3,k)of the related issues in McKay quiver.The McKay quiver is based on the Affine McKay quiver of type (?) which by adding a circle to each ver-tex.The properties of the Loewy matrix are discussed.The λ-matrix equiv-alent diagonal matrix D(l)and its related transformation matrix E(l)were obtained by elementary transformation of the A-matrix of three-dimensional McKay quiver of type (?).We get the primary factor of three-dimensional McKay quiver of type (?).Then we calculate the base vector of Jordan block in the Loewy matrix of eigenvalue 1.This article is based on[4]to consider the complexity of the linear combinations of these fundamental vectors.The conclusion is that the sum of the same complexity vectors is no more complex than the original complexity and the complexity of the different complexity vectors equals the larger complexity.