BRAUER CHARACTERS RELATIVE TO A NORMAL SUBGROUP

Abstract

Suppose that $N$ is a normal $p$-subgroup of a finite group $G$ and let $G^0$ be the set of elements of $G$ whose $p$-part lies in $N$. We prove the existence of a canonical basis ${\rm IBr}(G, N)$ of the space of complex class functions of $G$ defined on $G^0$, such that the restriction $\chi^0$ of any irreducible complex character $\chi$ of $G$ is a linear combination $\sum_{\phi\in{\rm IBr}(G, N)} d_{\chi \phi} \phi$ of the elements of this basis, where the $d_{\chi \phi}$ are non-negative integers. Furthermore, if we write $\Phi_\phi=\sum_{\chi} d_{\chi \phi}\chi$, then the $\Phi_\phi$ form the K\"ulshammer--Robinson ${\Bbb Z}$-basis of the ${\Bbb Z}$-module generated by the characters afforded by the $N$-projective $RG$-modules, where $R$ is a certain complete discrete valuation ring. By using these `decomposition numbers', it is possible to define a linking in the set of the irreducible complex characters of $G$. 1991 Mathematics Subject Classification: 20C15, 20C20.