On a conjecture of K. Harada. II
Let \(G\) be a finite group and \(G^ 0\) the set of \(p\)-regular elements of \(G\) for a prime \(p\). Let \(\{\chi_ 1,\dots,\chi_ n\}\) be the irreducible ordinary characters of \(G\) and put \(\rho_ J = \sum_{j \in J} \chi_ j(1) \chi_ j\) and \(\{\chi_ J\} = \{\chi_ j \mid j \in J\}\) for a subset \(J\) of the index set \(\{1,\dots,n\}\). \textit{K. Harada} [J. Algebra 70, 350-355 (1981; Zbl 0479.20007)] stated the following Conjecture A: If \(\rho_ J\) vanishes on \(G - G^ 0\), then \(\{\chi_ J\}\) is a union of \(p\)-blocks of \(G\). In Part I [cf. the preceding item Zbl 0793.20010] we show that Harada's conjecture holds for \(PSL(2,q^ f)\). In this paper, we prove the following theorems. Theorem 1. If \(G\) is the symplectic group \(\text{Sp}(4,q)\) where \(q\) is a power of an odd prime \(e\), then Conjecture A holds for every \(p\) different from \(e\). Theorem 2. If \(G\) is the finite Chevalley group \(G_ 2(q)\), where \(q\) is a power of a prime \(e\), then Conjecture A holds for every \(p\) different from \(e\).
scientific article; zbMATH DE number 562452