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 $\Lambda$. Let $\mathbb{Z}[\Lambda {]}^W$ and $S(\Lambda {)}^W$ be the $W$-invariant subrings of the integral group ring $\mathbb{Z}[\Lambda ]$ and the symmetric algebra $S(\Lambda )$ respectively. A celebrated result by Chevalley says that $\mathbb{Z}[\Lambda {]}^W$ is a polynomial ring in classes of fundamental representations ${\rho }_1,...,{\rho }_n$ and $S(\Lambda {)}^W\otimes Q$ is a polynomial ring in basic polynomial invariants $q_1,...,q_n$. In the present paper we establish and investigate the relationship between ${\rho }_i$'s and $q_i$'s over the integers. As an application we provide estimates for the torsion of the Grothendieck $\gamma$-filtration and the Chow groups of some twisted flag varieties up to codimension 4.