  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 symmetric
algebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that
$Z[M]^W$ is a polynomial ring over $Z$ in classes of fundamental
representations $w_1,...,w_n$ and $S^*(M)^{W}$ over rational numbers is a
polynomial ring in basic polynomial invariants $q_1,...,q_n$, where $n$ is the
rank. In the present paper we establish and investigate the relationship
between $w_i$'s and $q_i$'s over the integers. As an application we provide an
annihilator of the torsion part of the 3rd and the 4th quotients of the
Grothendieck gamma-filtration on the variety of Borel subgroups of the
associated linear algebraic group.
