Counting characters in blocks of solvable groups with abelian defect group

TL;DR

This paper investigates the conditions under which the number of lifts of Brauer characters in blocks of solvable groups with abelian defect groups reaches the known upper bound, linking local subgroup data to this phenomenon.

Contribution

It provides a necessary and sufficient condition for the maximum number of lifts to occur, based on local subgroup information, and applies this to the $k(B)$ conjecture.

Findings

Characterizes when the lift count reaches the upper bound

Links local subgroup data to lift counts in blocks

Analyzes cases of equality in the $k(B)$ conjecture

Abstract

If $G$ is a solvable group and $p$ is a prime, then the Fong-Swan theorem shows that given any irreducible Brauer character $\phi$ of $G$, there exists a character $\chi \in \irrg$ such that $\chi^o = \phi$, where $^o$ denotes the restriction of $\chi$ to the $p$-regular elements of $G$. We say that $\chi$ is a {\it{lift}} of $\phi$ in this case. It is known that if $\phi$ is in a block with abelian defect group $D$, then the number of lifts of $\phi$ is bounded above by $|D|$. In this paper we give a necessary and sufficient condition for this bound to be achieved, in terms of local information in a subgroup $V$ determined by the block $B$. We also apply these methods to examine the situation when equality occurs in the $k(B)$ conjecture for blocks of solvable groups with abelian defect group.