Cartan矩阵的分解和Brauer特征标次数猜想

Decomposition of Cartan matrix and conjectures on Brauer character degrees

设G为有限群,N是G的正规子群.记J=J(F[N])为F[N]的Jacobson根,I=Ann(J)={α∈F[G]|Jα=0}为J在F[G]中的零化子.本文主要研究了,根据F[G/N]和F[G]/I的Cartan矩阵,分解F[G]的Cartan矩阵.这种分解在Cartan不变量和G的合成因子之间建立了一些联系.本文指出N中p-亏零块的存在性依赖于Cartan不变量或者I在F[G]中的性质,证明了Cartan矩阵的分解部分地依赖于B所覆盖的N中的块的性质.本文研究了b为N上的块且l(b)=1时,覆盖b的G中的块B的性质.在两类情形下,本文证明了块代数上关于Brauer特征标次数的猜想成立,涵盖了Holm和Willems研究的某些情形.进而对Holm和Willems提出的问题给出了肯定的回答.另外,本文还给出了Cartan不变量的一些其它结果.

Abstract Let G be a finite group and N a normal subgroup of G. We denote by J = J（F[N]） the Jacobson radical of F[N] and by I = Ann（J） = {α∈ F[G] [Jα = 0} the annihilator of J in FIG]. In this paper, we study the decomposition of Cartan matrix of F[G] in terms of that of FIG[N] and F[G]/I. This decomposition establishes some connections between Cartan invariants and chief composition factors of G. We will prove that existing zero-defect p-blocks in N depend on the properties of I in F[G] or Cartan invariants. We shall demonstrate that the decomposition of Cartan matrix partly depends on properties of blocks in N covered by B. We are mainly concerned with the block B of G which covers a block b of N with l（b） = 1. In two cases, we will prove that the conjectures on Brauer character degrees hold for the block algebras, covering some cases studied by Holm and Willems. Furthermore we give an affirmative answer to the question raised by Holm and Willems in our cases. Some other results about Caftan invariants are presented here.