Basic polynomial invariants, fundamental representations and the chern class map

Abstract

Consider a crystallographic root system together with its Weyl group W acting on the weight lattice Λ. Let Z{double strok}[Λ]W and S(Λ)W be the W-invariant subrings of the integral group ring Z[Λ] and the symmetric algebra S(Λ) respectively. A celebrated result by Chevalley says that Z{double strok}[Λ]W is a polynomial ring in classes of fundamental representations ρ1, ..., ρn and S(Λ)W ⊗ Q{double struk} is a polynomial ring in basic polynomial invariants q1, ..., qn. In the present paper we establish and investigate the relationship between ρi's and qi's over the integers. As an application we provide estimates for the torsion of the Grothendieck γ-filtration and the Chow groups of some twisted flag varieties up to codimension 4.