The McKay-Slodowy Correspondence And Poincaré Series Of The Relative Groups

In 1980,J.McKay proposed a remarkable one-to-one correspondence between the isomorphism classes of the finite subgroups of the special linear group SL2(C)and the Dynkin diagrams of simply laced affine Lie algebras.In the same year,P.Slodowy discovered that the Dynkin diagrams of all affine Lie algebras can be realized by the fixed pairs of finite normal subgroups of SU2(C),which was known as the McKay-Slodowy correspondence.Since the introduction of McKay correspondence,it has promoted a deeper re-search of the relationship between groups and Lie algebras,in addition the applications have been found in combinatorics,algebraic geometry,representation theory and mathe-matical physics.Naturally,people are interested in the impact and application results of McKay-Slodowy correspondence on the relative mathematics branches.This disserta-tion will focus on the topic of the McKay-Slodowy correspondence and Poincare series of the relative groups.As a result,the following meaningful results were obtained:1.We use the restriction and inductive functors to provide a detailed exposition of the McKay-Slodowy correspondence from the perspective of group theory construc-tion,where we clarify some of the missing points in the literature of Slodowy.That is our treatment of types A2(2)and A2n(2)are new.In this way,the Dynkin diagrams of all affine Lie algebras are realized by the fixed pairs of groups.2.For arbitrary pairs of finite groups N<G,the general formulas of the Poincare series for the N-restriction of the irreducible G-modules and the induced modules of the irreducible N-modules in the tensor algebra T(V)=(?)k≥0 V(?)k are given using the general theory respectively.Moreover,the Poincare series mj(t)and mj(t)are explicitly computed in the tensor algebra T(V)for all the distinguished pairs of subgroups NΔG≤SU2(C).In particular,the Poincare series of tensor invariants provide a conceptual interpretation of the exponents of the affine Lie algebra in non-simply laced and twisted types.3.One goal of this dissertation is to generalize Kostant’s results in two directions.In the first direction,we consider the general special linear group SLn(C).In another aspect,we replace the defining fundamental module V=Cn by any fundamen-tal irreducible module of the Lie group SLn(C).For each pair of finite subgroups NΔG≤SLn(C),we give the formulas of the Poincare series of the N-restriction modules and induced G-modules in symmetric algebra S(Cn)=(?)k≥0 Sk(Cn)re-spectively.In particular,if N=G≤SLn(C)we get a general formula of the Poincare series for irreducible G-modules in symmetric algebra S(Cn).4.When N ≤G≤SL2(C),the Poincare series of symmetric invariants are obtained in terms of the quotient of the quantum finite Cartan matrices of finite dimensional Lie algebras and the quantum affine Cartan matrices of affine Lie algebras.In addi-tion,we generalize a classical result of Poincare series for symmetric G-invariants to Poincare series for G-restriction invariants or N-induction invariants.In other words,we have provided a unified formula of Poincare series for symmetric invari-ants for affine Lie algebras in both untwisted and twisted types.5.We also derive the Poincare series of symmetric invariants exclusively in terms of Tchebychev polynomials.This implies a surprising beautiful fact about the Poincare series of invariants that they are completely determined by the types of the distinguished pairs of subgroups and the respective dimensions of the subgroups.Our new formula points out the global picture of the Poincare series of invariants.In this sense,our method reveals some new feature that are only implicitly available so far in the literature.