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.