Counting characters in blocks of solvable groups with abelian defect group

Abstract

If $G$ is a solvable group and $p$ is a prime, then the Fong-Swan theorem shows that given any irreducible Brauer character $\varphi$ of $G$, there exists a character $χ\in \irrg$ such that ${\chi }^o=\varphi$, where ${}^o$ denotes the restriction of $\chi$ to the $p$-regular elements of $G$. We say that $\chi$ is a {\it{lift}} of $\varphi$ in this case. It is known that if $\varphi$ is in a block with abelian defect group $D$, then the number of lifts of $\varphi$ 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.