# Basic polynomial invariants, fundamental representations and the Chern class map

### Sanghoon Baek

Department of Mathematics Department of Mathematics and Statistics and Statistics University of Ottawa University of Ottawa### Erhard Neher

Department of Mathematics Department of Mathematics and Statistics and Statistics University of Ottawa University of Ottawa### Kirill Zainoulline

Department of Mathematics and Statistics University of Ottawa 585 King Edward Ottawa ON K1N 6N5 Canada

## Abstract

Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $Λ$. Let $Z[Λ]_{W}$ and $S(Λ)_{W}$ be the $W$-invariant subrings of the integral group ring $Z[Λ]$ and the symmetric algebra $S(Λ)$ respectively. A celebrated result by Chevalley says that $Z[Λ]_{W}$ is a polynomial ring in classes of fundamental representations $ρ_{1},...,ρ_{n}$ and $S(Λ)_{W}⊗Q$ is a polynomial ring in basic polynomial invariants $q_{1},...,q_{n}$. In the present paper we establish and investigate the relationship between $ρ_{i}$'s and $q_{i}$'s over the integers. As an application we provide estimates for the torsion of the Grothendieck $γ$-filtration and the Chow groups of some twisted flag varieties up to codimension 4.

## Cite this article

Sanghoon Baek, Erhard Neher, Kirill Zainoulline, Basic polynomial invariants, fundamental representations and the Chern class map. Doc. Math. 17 (2012), pp. 135–150

DOI 10.4171/DM/363