English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 thedecomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$.This decomposition establishs some connections between Cartan invariants andchief composition factors of $G$. We find that existing zero-defect $p$-blockin $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When weconsider the Cartan invariants for a block algebra $B$ of $G$, thedecomposition is related to what kind of blocks in $N$ covered by $B$. Wemainly 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 anaffirmative answer to a question by Holm and Willems in our cases. Some otherresults about Cartan invariants are presented in our paper.

Decomposition of Cartan Matrix and conjectures on Brauer character degrees