Brauer characters relative to a normal subgroup (Q2766358)

The paper under review concerns comparing representations of a finite group in characteristic \(p\) with those in characteristic zero. Let \(N\) be a normal \(p\)-subgroup of \(G\) and let \(G^0=\{x\in G\mid x_p\in N\}\). Let \(\text{cf}(G^-)\) denote the space of complex class functions of \(G\) defined on \(G^0\). If \(\chi\) is a complex irreducible character of \(G\) the restriction of \(\chi\) to \(G^0\) is denoted by \(\chi^0\). In one of the main results the author proves the existence of a canonical basis \(\text{IBr}(G,N)\) of \(\text{cf}(G^0)\) such that \(\chi\in\text{Irr}(G)\), then \(\chi^0=\sum_{\varphi\in\text{IBr}(G)}d_{\chi\varphi}\varphi\), where \(d_{\chi\varphi}\) are nonnegative integers.