[Paper Review] Cohomological invariants of finite Coxeter groups

Jérôme Ducoat|arXiv (Cornell University)|Dec 29, 2011

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper generalizes Serre's splitting principle for cohomological invariants to finite Coxeter groups over fields of characteristic zero, establishing a vanishing criterion based on restrictions to abelian subgroups generated by reflections. The key result is a complete description of the $ H^*(k_0, \mathbb{Z}/2\mathbb{Z}) $-module of cohomological invariants for Weyl groups of classical type, showing it is free with an explicit basis of products of Stiefel-Whitney invariants.

ABSTRACT

In this paper, we generalize Serre's splitting theorem for cohomological invariants of the symmetric group to finite Coxeter groups, provided that the ground field has characteristic zero. We then use this principle to determine all the cohomological invariants of Weyl groups of classical type with coefficients modulo 2.

Motivation & Objective

• To extend Serre’s splitting principle for cohomological invariants of symmetric groups to finite Coxeter groups over fields of characteristic zero.
• To characterize the structure of cohomological invariants for Weyl groups of classical types with coefficients in $ \mathbb{Z}/2\mathbb{Z} $.
• To establish a vanishing criterion for invariants based on their restrictions to abelian subgroups generated by reflections.
• To determine a free basis for the module of cohomological invariants of Weyl groups of type $ B_n $, $ C_n $, and $ D_n $ over $ H^*(k_0, \mathbb{Z}/2\mathbb{Z}) $.

Proposed method

• Generalizes Serre’s splitting principle by proving that if a cohomological invariant vanishes on all abelian subgroups generated by reflections, then it vanishes globally, under characteristic zero assumptions.
• Uses Galois descent and the structure of étale algebras to relate invariants of Weyl groups to those of their maximal abelian subgroups generated by reflections.
• Introduces explicit invariants $ a_{r,s} $ on the maximal abelian subgroup $ H_m $ of $ W $, which are fixed under the action of its normalizer.
• Applies restriction maps to relate invariants of the full Weyl group $ W $ to those of $ H_m $, proving injectivity of the restriction map.
• Constructs a basis of the invariant module using products of Stiefel-Whitney invariants $ w_i $ and $ \widetilde{w}_j $, with $ j $ even.
• Employs combinatorial identities and linear algebra over $ H^*(k_0, \mathbb{Z}/2\mathbb{Z}) $ to show that the image of the restriction map spans the full invariant module of $ H_m $.

Experimental results

Research questions

• RQ1 Can Serre’s splitting principle for symmetric groups be extended to arbitrary finite Coxeter groups over fields of characteristic zero?
• RQ2 What is the structure of the module of cohomological invariants for Weyl groups of classical type with coefficients in $ \mathbb{Z}/2\mathbb{Z} $?
• RQ3 How do the restrictions of invariants to abelian subgroups generated by reflections determine the global invariants?
• RQ4 What is a free basis for the $ H^*(k_0, \mathbb{Z}/2\mathbb{Z}) $-module of cohomological invariants of Weyl groups of type $ B_n $, $ C_n $, and $ D_n $?

Key findings

• The restriction map $ \text{Res}_W^{H_m} : \text{Inv}_{k_0}(W, \mathbb{Z}/2\mathbb{Z}) \to \text{Inv}_{k_0}(H_m, \mathbb{Z}/2\mathbb{Z})^{N_{H_m}/H_m} $ is injective for Weyl groups of classical type.
• The module $ \text{Inv}_{k_0}(H_m, \mathbb{Z}/2\mathbb{Z})^{N_{H_m}/H_m} $ is free over $ H^*(k_0, \mathbb{Z}/2\mathbb{Z}) $ with basis $ \{a_{r,s}\} $, where $ r,s \geq 0 $, $ r+s \leq n/2 $.
• The $ H^*(k_0, \mathbb{Z}/2\mathbb{Z}) $-module $ \text{Inv}_{k_0}(W, \mathbb{Z}/2\mathbb{Z}) $ is free with basis $ \{w_i \cdot \widetilde{w}_j\} $, where $ 0 \leq i \leq \lfloor n/2 \rfloor $, $ 0 \leq j \leq 2(\lfloor n/2 \rfloor - i) $, and $ j $ even.
• The invariants $ w_i $ and $ \widetilde{w}_j $ arise from the Stiefel-Whitney classes of the standard quadratic forms associated to étale algebras and their twistings.
• The restriction of $ w_i \cdot \widetilde{w}_j $ to $ H_m $ is a linear combination of the $ a_{r,s} $, and the coefficients are binomial coefficients.
• An inductive argument shows that every $ a_{r,s} $ lies in the image of the restriction map, proving surjectivity and hence a complete basis is obtained.