Counting characters in blocks of solvable groups with abelian defect group

Details Counting characters in blocks of solvable groups with abelian defect group Counting characters in blocks of solvable groups with abelian defect group

2011-12-16

arxiv.org/abs/1112.3819v1

Mathematics

Group Theory

Scientific paper

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.

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