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

TL;DR

This paper establishes a relationship between the p-part of the conductor of a generalized character and its decomposition numbers, with implications for block theory and equivalences in modular representation theory.

Contribution

It proves the equality of the p-part of the conductor and the conductor of decomposition numbers, and shows preservation under certain isotypies and isometries.

Findings

p-part of the conductor equals the conductor of decomposition numbers

p-parts of conductors are preserved under specific equivalences

applies to blocks with abelian defect and Frobenius inertial quotient

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.