Witt invariants of Weyl groups

TL;DR

This paper characterizes the Witt invariants of Weyl groups over a field, providing generators for the module of invariants and extending known results to additional types, with computational methods applicable to broader algebraic groups.

Contribution

It introduces a comprehensive description of Witt invariants for Weyl groups, extending previous results and developing computational techniques for general algebraic groups.

Findings

Witt invariants of Weyl groups are generated by trace forms and exterior powers for types B, C, D, and G2.

The paper extends Serre's results from type A to other Weyl group types.

Develops methods for lifting invariants from cohomological to Witt invariants.

Abstract

We describe the Witt invariants of a Weyl group over a field $k_0$ by giving generators for the $W(k_0)$-module of Witt invariants, under the assumption that the characteristic of $k_0$ does not divide the order of the group. For the Weyl groups of types $B_n$, $C_n$, $D_n$, and $G_2$, we show that the Witt invariants are generated as a $W(k_0)$-algebra by trace forms and their exterior powers, extending a result due to Serre in type $A_n$. Many of our computational methods are applicable to computing Witt invariants of any smooth linear algebraic group over $k_0$, including a technique for lifting module generators from cohomological invariants to Witt invariants.