Chern classes for representations of reductive groups

TL;DR

This paper computes the graded ring associated with the representation ring of a complex reductive group and introduces a straightforward method to define and compute Chern classes of principal bundles and their associated vector bundles.

Contribution

It provides the first explicit computation of the graded ring of the representation ring and simplifies the definition and calculation of characteristic classes for principal and vector bundles.

Findings

Computed the graded ring gr R(G) over Q for complex reductive groups.

Provided a simple definition of characteristic classes in gr K(B).

Offered an easy method to compute Chern classes of associated vector bundles.

Abstract

Let G be a complex connected reductive group. The representation ring R(G) admits a canonical filtration defined in terms of the lambda-structure. We compute the associated graded ring gr R(G) (over Q) and the Chern classes of a representation -- an easy exercise which we have been unable to find in the literature. As an application we give a simple definition of the characteristic classes of a principal bundle P --> B in gr K(B), and a simple way of computing the Chern classes of the associated vector bundles.