Result: Minimality of an Active Fragment of a Character Table of a Finite Group: Minimality of an active fragment of the character table of a finite group

Let \(G\) be a finite group, let \(D\) be a normal subset of \(G\), and let \(\Phi\subseteq\text{Irr}(G)\) be a set of characters of \(G\). Given \(\varphi\in\Phi\), define the cut-off function \(\varphi|^0_D\) by the formula \(\varphi|^0_D(x)=x\) for \(x\in D\) and \(\varphi|^0_D(x)=0\) for \(x\in G\setminus D\). Say that \(D\) interacts with \(\Phi\) if, for every \(\varphi\in\Phi\), its cut-off function \(\varphi|^0_D\) is a linear combination of characters in \(\Phi\). Let \(X\) be the whole character table of the group and let \(X(\Phi,D)\) be the submatrix of the values of characters in \(\Phi\) on elements of \(D\). If \(D\) interacts with \(\Phi\) then the submatrix is called an active fragment of the table \(X\). For an arbitrary finite group \(G\) the author finds conditions on \(\Phi\) and \(D\) to compose an active fragment of the table of characters.