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Character triples and weights

Zhicheng Feng

TL;DR

This work develops a unified framework linking character triples, block theory, and weights through a novel relation built on associated projective representations. It generalizes central and block isomorphisms to a broader setting, provides concrete criteria and constructions (including direct and wreath products) to propagate these relations, and integrates Dade's ramification group to control block behavior across subgroups. The paper also establishes a Clifford- theory–driven program for weights, including covering, DGN correspondences, and going-up properties, enabling equivariant bijections between irreducible characters and weights in a way that supports inductive proofs of local-global conjectures. The techniques are particularly aimed at applications to the Alperin weight conjecture and representations of groups of Lie type at good primes, offering new tools for transferring local data to the global group structure.

Abstract

We define a new relation between character triples and prove some Clifford theory properties for weights in terms of character triples.

Character triples and weights

TL;DR

This work develops a unified framework linking character triples, block theory, and weights through a novel relation built on associated projective representations. It generalizes central and block isomorphisms to a broader setting, provides concrete criteria and constructions (including direct and wreath products) to propagate these relations, and integrates Dade's ramification group to control block behavior across subgroups. The paper also establishes a Clifford- theory–driven program for weights, including covering, DGN correspondences, and going-up properties, enabling equivariant bijections between irreducible characters and weights in a way that supports inductive proofs of local-global conjectures. The techniques are particularly aimed at applications to the Alperin weight conjecture and representations of groups of Lie type at good primes, offering new tools for transferring local data to the global group structure.

Abstract

We define a new relation between character triples and prove some Clifford theory properties for weights in terms of character triples.

Paper Structure

This paper contains 14 sections, 21 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Induced blocks
  3. Notation
  4. Induced blocks
  5. Dade's ramification group
  6. A new relation between character triples
  7. Character triples and projective representations
  8. A generalization of central and block isomorphisms
  9. A criterion for $\geqslant_{(g),b}$
  10. Construction of $\geqslant_{(g),c}$ and $\geqslant_{(g),b}$
  11. Weights and Clifford theory
  12. Covering of weights
  13. A criterion to establish $\geqslant_{(g),b}$
  14. A going-up property

Key Result

Lemma 2.1

Let $N\unlhd G$, $H\le G$ and $M=H\cap N$. Suppose that there exist characters $\widetilde{\theta}\in\operatorname{Irr}(G)$ and $\widetilde{\varphi}\in\operatorname{Irr}(H)$ with $\operatorname{Res}^G_{N}(\widetilde{\theta})\in\operatorname{Irr}(N)$ and $\operatorname{Res}^H_M(\widetilde{\varphi})\i

Theorems & Definitions (51)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.7
  • proof
  • Remark 3.1
  • Definition 3.2
  • ...and 41 more