Characteristic Classes Of Representations Of Lie Groups
Rohit Joshi, Steven Spallone
TL;DR
The paper develops a unified, computation-friendly framework to express characteristic classes of representations of complex reductive Lie groups in terms of torus data. By introducing the power-sum polynomials $P_k$ of weights and the auxiliary antisymmetric polynomials $F_k$, it converts the Weyl character formula into a recursive procedure for $P_k(\phi_\lambda)$, hence obtaining polynomial expressions for torus-restricted Chern classes $c_k^T(\pi_\lambda)$ and Stiefel-Whitney classes $w_k^T(\pi_\lambda)$ as functions of the highest weight $\lambda$. The authors develop explicit invariant-theoretic machinery to compute higher $F_k$ and provide detailed examples for $\mathfrak{sl}_2$, $\mathfrak{sl}_3$, and classical groups, including isogeny considerations and a spinoriality criterion. A parallel treatment of Stiefel-Whitney classes via restriction to $T[2]$ yields factorization formulas and concrete computations for $\mathrm{SL}_2$ and $\mathrm{SL}_3$, together with an analogue for total Chern classes. The work yields practical tools for lifting representations to spin groups and clarifies when torus-restrictions determine global characteristic classes, with broader implications for representation theory and topology of classifying spaces.
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.