Some Research On Chevalley-Shephard-Todd’s Theorem In Modular Invariant Theory

In nonmodular invariant theory,Chevalley-Shephard-Todd’s theorem is one of the core results.It determines which groups have the polynomial rings of invariants.In this paper,we prove some conjectures related to Chevalley-Shephard-Todd theorem in modular invariant theory.Concretely,the main contents are as follows:At first,this paper proves Kemper and Malle’s conjecture for the transvection free groups with polynomial rings of invariants.We characterize transvection free groups by the classifica-tion of finite irreducible pseudoreflection groups and analysis the rings of invariants.Finally,we extend Kemper and Malle’s result for the irreducible transvection free groups to the reducible case,and show that the ring of invariants of transvection free group is polynomial if and only if the pointwise stabilizer of any subspace is generated by pseudoreflections.Subsequently this paper gives a new method of the construction of modular invariants.We use the new operator to construct modular invariants instead of the transfer and show that it behaves better.Broer’s conjecture is another description of Chevalley-Shephard-Todd theorem in modular invariant theory,so we also confirm Broer’s conjecture for the transvection free groups.Finally,this paper describes the variety defined by the twisted transfer ideal.It turns out that this variety is nothing but the union of reflecting hyperplanes and the fixed subspaces of some specific elements of the group.