Finite-dimensional algebras of integral $p$-adic representations of finite groups

P. M. Gudivok ; S. F. Goncharova ; V. P. Rud'ko

Sbornik. Mathematics, Tome 23 (1974) no. 3, pp. 336-361

Résumé

Let $F$ be a inite extension of the field of rational $p$-adic numbers $Q_p$, $R$ he ring of integers of $F$, $G$ a finite group, $a(RG)$ the ring of $R$-representations of $G$ and $A(RG)=Q\otimes_Za(RG)$ ($Z$ is the ring of rational integers and $Q$ the rational number field). We study the algebra $A(RG)$ in the case where the number $n(RG)$ of indecomposable $R$-representations of $G$ is finite. In particular, for $G$ a $p$-group and $n(RG)\infty$ we find a list of the tensor products of indecomposable $R$-representations of $G$ and obtain a description of the radical $N$ of $A(RG)$ and of the quotient algebra $A(RG)/N$. It turns out that in this case we always have $N^2=0$. Bibliography: 26 titles.