On the decomposability of mod 2 cohomological invariants of Weyl groups

TL;DR

This paper analyzes mod 2 cohomological invariants of Weyl groups, showing they decompose into sums induced by Stiefel-Whitney classes and Witt ring invariants, using Serre's splitting principle for detection.

Contribution

It provides a decomposition of mod 2 invariants of Weyl groups into elementary components, extending understanding of their structure over fields with characteristic coprime to the group order.

Findings

Invariants decompose as direct sums of coefficient modules.

Basis elements are induced by Stiefel-Whitney classes or Witt ring invariants.

Detection of invariants is guaranteed on elementary abelian 2-subgroups.

Abstract

We compute the invariants so 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.