Basic Polynomial Invariants, Fundamental Representations and the Chern Class Map

Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $\Lambda$. Let $\ZZ[\Lambda]^W$ and $S(\Lambda)^W$ be the $W$-invariant subrings of the integral group ring $\ZZ[\Lambda]$ and the symmetric algebra $S(\Lambda)$ respectively. A celebrated result by Chevalley says that $\ZZ[\Lambda]^W$ 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.

2010 Mathematics Subject Classification: Primary 13A50; Secondary 14L24

Keywords and Phrases: Dynkin index, polynomial invariant, fundamental representation, Chow group, gamma-filtration, twisted flag variety, torsion

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