Decomposition matrices over unitriangulars (Q2682830)

Let \(G\) be a finite group and \(p\) a prime. If \(\mathrm{Irr}(G)\) is the set of the irreducible complex characters of \(G\), and \(\mathrm{IBr}(G)\) is a set of irreducible \(p\)-Brauer (or modular) characters of \(G\), then the decomposition numbers \(d_{\chi\varphi}\) can be calculated using the equation \(\chi^{0}=\sum_{\varphi \in \mathrm{IBr}(G)} d_{\chi \varphi}\, \varphi\), where \(\chi^{0}\) is the restriction of \(\chi\) to the set of elements of \(G\) whose order is not divisible by \(p\). The matrix \(D \big ( d_{\chi \varphi} \big )\) is the \textit{decomposition matrix}, it codifies the main relationships between the complex and modular irreducible representations of \(G\). Let \(X \subseteq \mathrm{Irr}(G)\) and \(Y \subseteq \mathrm{IBr}(G)\), then \(D_{X,Y}\) is the submatrix of \(D\) indexed by the rows in \(X\) and the columns in \(Y\) (subject to some ordering). The matrix \(D_{X,Y}\) is called \textit{unitriangular} (respectively \textit{unimodular}) if \(|X|=|Y|\) and, for some ordering, \(D_{X,Y}\) is square lower unitriangular (respectively unimodular). A first result proved in this paper is Theorem A: Let \(G\) be a finite group and \(N \trianglelefteq G\). Suppose that \(B \subseteq \mathrm{Irr}(N)\) and \(C \subseteq \mathrm{IBr}(N)\) are \(G\)-stable sets such that \(D_{B,C}\) is unitriangular. If the Sylow \(p\)-subgroups of \(G/N\) are cyclic or generalized quaternion, then there is \(\widetilde{B} \subseteq \mathrm{Irr}(G|B)\) such that \(D_{\widetilde{B}, \mathrm{IBr}(G|C)}\) is unitriangular. Theorem B is a generalization of Theorem A whose statement is too technical to be reported here.