Study On The Properties Of Loewy Matrix For Three-dimensional McKay Quiver Of The Type ?D₄|^?

In 1980,McKay introduces the concept of McKay quiver.He also observes that the McKay arrow of the finite subgroup G of SL(2,k)is an affine Dynkin graphs (?),and classical McKay correspondence asserts that there exists one to one correspondence between irreducible representations of finite subgroup G of SL(2,k)and the cohomology of minimal resolution of Klein singularities C~2/G.When the finite subgroup G of SL(m,k)is embedded in SL(m+1,k),the new McKay quiver is obtained by adding a circle at each vertex of the original McKay quiver.This article discusses the vectors of different complexity related to the Loewy matrix for three-dimensional McKay quiver of (?).We calculated the Jordan block and the corresponding base vector when the eigenvalue of the Loewy matrix for three-dimensional McKay quiver of (?) is 1.We portrayed the vector cases of different complexity related to the Loewy matrix by the base vector of the Jordan block.We have obtained that the nonnegative vector related to the Loewy matrix for three-dimensional McKay quiver of (?) is 1,2,3.And then we also get the complexity of the eigenvectors of 1 when the eigenvalues are 1.The complexity of the vector in the space L that is made up of the base vector of the Jordan block ? 3.If the complexity of the vector in space L is d,then the complexity of the sum or difference of these vectors? d.And the complexity of the sum of the vectors of different complexity in the space L is a larger complexity.