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

In 1980,McKay[18]introduced the concept of McKay quiver.He also ob-serves that,for a finite subgroup G(?)SL(2,k),the McKay quiver QG is one of the extended Dynkin graphs An,Dn,(?),E7,E8.Guo[13]has recently point,ed out taht,the McKay quiver of a finite subgroup G of GL(m,k)is a regular covering of the McKay quiver of its normal subgroup G G∩SL(m,k),and when embedding G in a canonical way into SL(m + 1,k),the new McKay quiver is obtained by adding an arrow from the Nakayama translation of i back to i for each vertex i.In this paper,we study the McKay quiver which is based on the reconstructed quiver constructed by the Affine Dynkin graphs (?),and then adds a circle at each vertex.Complexity as a kind of important numerical invariants of said,in the s-tudy of group presentation and algebraic representation is very important.This paper mainly discusses the properties of Loewy matrix for three-dimensional McKay quiver of the type (?),depicts the vector of different complexity of the Loewy matrix:first,of all,we will get the Loewy matrix for three-dimensional McKay quiver of the type (?) initial transformation by using diagonal matrix D(l)and the corresponding transformation matrix E(l),then,the elementary factors of the Loewy matrix are obtained,if the block and the basis vectors,and then studied its characteristic root of 1 Loewy matrix,or.Jordan blocks corresponding to the complexity of the linear combination of the base vector proved the characteristic root for 1 feature subspace is the complexity of 1 to quantum space;and the additive property of complexity is discussed:the complexity of the sum(difference)of the same complexity vector is not greater than the original complexity;the complexity of the sum of the different com-plexity vectors equals the greater complexity.