Cartan矩阵,分解数与单模的顶

Abstract

设Ｂ是有限群Ｇ的ｐ－块代数，考虑以Ｂ的Ｂｒａｕｅｒ不可约特征标为基底生成的自由Ａｂｅｌ群，以及由Ｂ的Ｃａｒｔａｎ矩阵决定的这个群的商群．通过研究这个商群中元素的阶数，我们得到了Ｃａｒｔａｎ矩阵和Ｃａｒｔａｎ数与单模的顶点及亏群的一些新关系式．对于常不可约特征标的高度， 我们得到了它与分解数以及Ｃａｒｔａｎ不变量的一些新关系式．

Let B be a p-block of a finite group G. We consider a free Abelian group with a basis of Brauer irreducible characters of the block B, and its factor group determined by Cartan matrix of the block B . Let C=(cij) and D=(dij) be the Cartan matrix and the decomposition matrix of B, respectively. We give some new results about the relations among C, defect group of B and the vertices of simple modules. We also get some new results about the decomposition numbers dij, the heights of ordinary irreducible characters of B and Cartan invariants.