Poincare Series For Tensor Invariants Of Abelian Group H_m,n

By using Benkart’s general results on Poincare series of invariants,we mainly studied the abelian subgroup H_m,n of SL3（C）especially the Poincare series of’ tensor invariants of H_m,n Furthermore,we obtain some general results on the Poincare series of tensor invariants of H1,nIn the introduction,we introduce the research background of Poincare series and McKay correspondence and a brief introduction of our work.In chapter two,we recall some preliminary knowledge on Lie algebras,including M-cKay correspondence,tensor algebra,tensor invariants,representation graph and Poincar-e Series.Moreover,we recall Benkart’s results and the results on classification of finite subgroups of SL3（C）.Chapter three is the main part of this paper,we compute the character of group H_m,n)and the adjoint matrix of the representation graph corresponding to the natural representation of H_m,n Furthermore,we verified that the natural representation V = C3 of H_m,n satisfies the conditions of Benkart’s Theorem,and compute some Poincare Series of tensor invariants of H_m,n.Finally,we give a general result on the Poincare series of’tensor invariants for H1,n.