On the decomposability of mod 2 cohomological invariants of Weyl groups

Abstract

We compute the invariants of Weyl groups in mod 2 Milnor $K$-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order, the invariants decompose as direct sums of the coefficient module. All basis elements are induced either by Stiefel–Whitney classes or specific invariants in the Witt ring. The proof is based on Serre’s splitting principle that guarantees detection of invariants on elementary abelian 2-subgroups generated by reflections.