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Consider a crystallographic root system together with its Weyl group $W$acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the$W$-invariant subrings of the integral group ring $Z[M]$ and the symmetricalgebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that$Z[M]^W$ is a polynomial ring over $Z$ in classes of fundamentalrepresentations $w_1,...,w_n$ and $S^*(M)^{W}$ over rational numbers is apolynomial ring in basic polynomial invariants $q_1,...,q_n$, where $n$ is therank. In the present paper we establish and investigate the relationshipbetween $w_i$'s and $q_i$'s over the integers. As an application we provide anannihilator of the torsion part of the 3rd and the 4th quotients of theGrothendieck gamma-filtration on the variety of Borel subgroups of theassociated linear algebraic group.

Basic polynomial invariants, fundamental representations and the Chern class map