Decomposition of Cartan Matrix and conjectures on Brauer character degrees

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dc.creatorJiwen, Zeng
dc.date2007-07-24
dc.date2008-10-20
dc.date.accessioned2026-07-07T10:11:07Z
dc.date.available2026-07-07T10:11:07Z
dc.descriptionLet $G$ 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 $I={\rm Ann}(J)=\{α\in F[G]|Jα=0\}$. 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.
dc.description16 pages, changed contents and title
dc.identifierhttps://arxiv.org/abs/0707.3495
dc.identifierhttp://arxiv.org/abs/0707.3495
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/171764
dc.subjectGroup Theory
dc.subjectRepresentation Theory
dc.subject20c20
dc.titleDecomposition of Cartan Matrix and conjectures on Brauer character degrees
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