[Paper Review] Chern classes for representations of reductive groups

Arnaud Beauville|ArXiv.org|Apr 3, 2001

Advanced Algebra and Geometry3 references3 citations

TL;DR

This paper establishes a canonical isomorphism between the rational graded representation ring of a reductive group G and the ring of invariant polynomial functions on its Lie algebra, showing that Chern classes of representations correspond to determinants of 1 + Lρ and Chern characters to traces of exp(Lρ), providing a purely algebraic, cohomology-free method to compute characteristic classes of associated vector bundles via invariant polynomials.

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.

Motivation & Objective

• To provide a purely algebraic, cohomology-free definition of Chern classes for representations of reductive groups.
• To establish a canonical isomorphism between the rational graded representation ring R(G) and the ring of invariant polynomial functions on the Lie algebra g.
• To show that Chern classes and Chern characters of representations are given by det(1 + Lρ) and Tr(exp(Lρ)) respectively, where Lρ is the Lie algebra representation.
• To apply the results to principal G-bundles, defining characteristic classes in the graded K-theory of the base space.

Proposed method

• Uses the λ-ring structure on the representation ring R(G), with λ-operations induced by exterior powers of representations.
• Applies Grothendieck's γ-filtration and Chern class construction via γp(x − ε(x)) in the associated graded ring.
• Leverages the fact that R(T), for a maximal torus T, is a split λ-ring, enabling explicit computation of the γ-filtration.
• Shows that the γ-filtration on R(G) is induced from that on R(T) via Weyl group invariance and the behavior of Adams operations under the γ-filtration.
• Uses the isomorphism between Pol(g)^inv and the graded representation ring, induced by restriction to a Cartan subalgebra h.
• Applies the characteristic homomorphism c_P: Pol(g)^inv → gr_Q K(B) for a principal G-bundle P over B, mapping invariant polynomials to characteristic classes in the graded K-theory of B.

Experimental results

Research questions

• RQ1 How can Chern classes for representations of reductive groups be defined without reference to cohomology or Chow rings?
• RQ2 What is the precise algebraic structure of the graded representation ring gr_Q R(G) for a reductive group G?
• RQ3 How do the Chern classes and Chern character of a representation ρ ∈ R(G) relate to the Lie algebra representation Lρ?
• RQ4 Can characteristic classes of associated vector bundles be computed algebraically from invariant polynomials on the Lie algebra?
• RQ5 What is the relationship between the characteristic classes of a principal G-bundle P and the Chern classes of the associated vector bundle P^ρ?

Key findings

• The rational graded representation ring gr_Q R(G) is canonically isomorphic to the ring of invariant polynomial functions on the Lie algebra, Pol(g)^inv.
• The total Chern class c(ρ) of a representation ρ is equal to det(1 + Lρ) as an invariant polynomial on g.
• The Chern character ch(ρ) is equal to Tr(exp(Lρ)) in the ring of invariant polynomial functions on g.
• For a principal G-bundle P over a base B, the characteristic classes c_P^{(i)} in gr_Q K(B) are obtained by evaluating the invariant polynomials I_i on the Lie algebra representation.
• The total Chern class of the associated vector bundle P^ρ is F(c_P^{(1)}, ..., c_P^{(ℓ)}) where F is the polynomial such that det(1 + Lρ) = F(I_1, ..., I_ℓ).
• For orthogonal and symplectic groups, the Chern classes c_P^{(p)} correspond to c_{2p}(E_P), and for SO(2ℓ), the last class c_P^{(ℓ)} satisfies (c_P^{(ℓ)})^2 = c_{2ℓ}(E_P).