The Structure Of The Mckay Quiver

It is well known that both the Dynkin diagrams An,Dn,E6,E7,E8 and the extended Dynkin ones An,Dn,E6,E7,Eg occur in many branches in mathematics, and they play an important role in lots of classification theorems. And the McKay quivers are such a quiver connected tightly with these diagrams.In 1980, J.McKay observed that their underlying graphs are one of the extended Dynkin diagrams for these McKay quivers of the finite subgroups G c SL(2,C). These diagrams occur respectively for the conjugacy classes of these subgroups. Naturally, we can ask, in general, for the McKay quiver of G which is a finite subgroup of a generalized linear group, what its construction is and what relation there is with the conjugacy class of the group G.In this paper, using two algebras associated to the McKay quiver Q: the Koszul graded selfinjective algebra A(Q) and its Yoneda one F(Q), we mainly discuss some properties of McKay quivers, and we also present a criterion that a quiver is not a McKay quiver in a particular case. These are necessary to acquaint ourselves with the construction of McKay quivers. Now we lay out our main results as follows:Proposition 3.4 Let Q be the McKay quiver of a finite subgroup G GL(m,C) = GL(V), = Q + } +..... + m and A;(i = 1,.....,m) the layermatrices of Q. Then AsAt >As+t for any l 3 . And suppose that F is not a Noetherian algebra, L, L are the Loewy matrices of A, A respectively and K is a Koszul cone of A which is invariant with the action of the matrix L - L . Then is also not Noetherian.Corollary 4.12 Let A be a Koszul graded selfinjective algebra, its Yoneda algebra and not Noetherian, Q and K its quiver and Koszul cone respectively, and ll(A) = 4 . And suppose that Q ^ Q is another quiver which satisfies Q0 = Q0, A, A are the matrices of Q,Q respectively. If either of the following conditions is satisfied, then Q is definitely not a McKay quiver of any finite subgroup of SL(3,C):(a) K is an invariant cone left by the matrixon(b) There is a basis in K of R; such that the matrix of a certain basis is a non-negative one.