Characteristic Classes Of Representations Of Lie Groups

TL;DR

This paper develops methods to compute characteristic classes of irreducible representations of reductive Lie groups, expressing them as polynomial functions of highest weights, linking algebraic and topological invariants.

Contribution

It introduces a procedure to compute symmetric polynomials of weights and expresses characteristic classes as functions of highest weights for reductive Lie groups.

Findings

Computed symmetric polynomials of weights as functions of highest weight

Expressed Chern classes of representations as polynomial functions of highest weight

Derived formulas for Stiefel-Whitney classes of orthogonal representations

Abstract

An irreducible representation of a reductive Lie algebra, when restricted to a Cartan subalgebra, decomposes into weights with multiplicity. The first part of this paper outlines a procedure to compute symmetric polynomials (e.g., power sums) of this multiset of weights, as functions of the highest weight. Next, let G be a connected reductive complex algebraic group with maximal torus T. We express the restrictions of the Chern classes of irreducible representations of G to T, as polynomial functions in the highest weight. We do the same for Stiefel-Whitney classes of orthogonal representations.