Decomposition of Cartan Matrix and conjectures on Brauer character degrees

TL;DR

This paper investigates the decomposition of Cartan matrices in group algebras related to a finite group and its normal subgroup, establishing connections with block theory, Cartan invariants, and conjectures on Brauer character degrees.

Contribution

It introduces a new decomposition approach for Cartan matrices linked to normal subgroups and proves Willems' conjecture in specific cases, also addressing a question by Holm and Willems.

Findings

Decomposition of Cartan matrices relates to chief factors of G.

Willems' conjecture holds for certain blocks covering a block with l(b)=1.

Affirmative answer to Holm and Willems' question in specific cases.

Abstract

Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let $J=J(F[N])$ denote the Jacboson radical of $F[N]$ and $I={\rm Ann}(J)=\{\alpha \in F[G]|J\alpha =0\}$. We have another algebra $F[G]/I$. We study the decomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$. This decomposition establishs some connections between Cartan invariants and chief composition factors of $G$. We find that existing zero-defect $p$-block in $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When we consider the Cartan invariants for a block algebra $B$ of $G$, the decomposition is related to what kind of blocks in $N$ covered by $B$. We mainly consider a block $B$ of $G$ which covers a block $b$ of $N$ with $l(b)=1$. In two cases, we prove Willems' conjecture holds for these blocks, which covers some true cases by Holm and Willems. Furthermore We give an affirmative answer to a question by Holm and Willems in our cases. Some other results about Cartan invariants are presented in our paper.