Partial inner products.
\(G\) is a finite group, \(S\) is a subset of \(\text{Irr}(G)\). For any class functions \(\alpha\) and \(\beta\) of \(G\), \(\sum_{\sigma\in S}[\alpha,\sigma][\sigma,\beta]\) is called the \(S\)-partial inner product of \(\alpha\) and \(\beta\). Assume characters and class functions are over \(\mathbb{C}\) and \(\chi\mapsto\overline\chi\) is the complex conjugation. For any \(g\) and \(h\) in different conjugacy classes of \(G\), the orthogonal property is well known: \(\sum_{\chi\in\text{Irr}(G)}\chi(g)\overline{\chi(h)}=0\). A weak second orthogonal property is introduced in a Clifford setting. That is, for any normal subgroup \(N\) of \(G\), and \(\varphi\in\text{Irr}(N)\), and \(\text{Irr}(G|\varphi)\) is the set of irreducible constituents of \(\varphi^G\), then \(\sum_{\chi\in\text{Irr}(G)}\chi(g)\overline{\chi(h)}=0\), with \(\overline g\) and \(\overline h\) not conjugate in \(\overline G=G/N\). The paper also looks into the Burnside-Brauer theorem for two nonfaithful characters \(\eta,\delta\) and the exponents \(\delta\eta^i\), and proves that those which are constituents of the exponents are exactly those with nonzero \([\chi|_K,\delta|_K]\), where \(K\) is the kernel of \(\eta\). That is, for any two characters \(\gamma\) and \(\delta\) of \(G\) with class function \(\eta\) such that the kernel \(K\) of \(\eta\) is \(G(\eta,\eta(1))\), the following are equivalent: (i) \([\gamma|_K,\delta|_K]\neq 0\); (ii) \([\gamma,\delta\eta^i]\neq 0\) for some \(i<m\); (iii) \([\gamma,\delta\eta^i]\neq 0\) for some \(i\), where \(m\) is the number of different values of \(\eta\).
scientific article; zbMATH DE number 5078826