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On the p-part of the conductor of a generalised character

Markus Linckelmann

Abstract

We show that the $p$-part of the conductor of a generalised character of a finite group is equal to the conductor of its generalised decomposition numbers. We use this to show that $p$-parts of conductors of irreducible characters are preserved under isotypies and perfect isometries that arise in the context of stable equivalences of Morita type with endopermutation source. We apply this to blocks with abelian defect and Frobenius inertial quotient.

On the p-part of the conductor of a generalised character

Abstract

We show that the -part of the conductor of a generalised character of a finite group is equal to the conductor of its generalised decomposition numbers. We use this to show that -parts of conductors of irreducible characters are preserved under isotypies and perfect isometries that arise in the context of stable equivalences of Morita type with endopermutation source. We apply this to blocks with abelian defect and Frobenius inertial quotient.

Paper Structure

This paper contains 6 sections, 23 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']} and Theorem \ref{['thm15']}
  3. On generalised decomposition numbers
  4. Quoted results on endopermutation modules
  5. Proof of Theorem \ref{['thm2']}
  6. Some remarks on conductors and restriction to blocks of subgroups

Key Result

Theorem 1.1

Let $G$ be a finite group. Assume that $K$ contains a primitive $|G|$-th root of unity. For any $\chi\in$${\mathbb Z}\mathrm{Irr}(G)$ we have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark 1.8
  • Remark 1.9
  • proof : Proof of Theorem \ref{['thm1']}
  • ...and 40 more