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 $M$. 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.