Group characters, permutation actions and sharpness (Q1867282)

If \(L\) is a set of complex numbers, let \(f_L(x)\) be the monic polynomial whose roots are \(L\) (with multiplicity \(1\)). Cameron and Kiyota observed that one could generalize an old result of Blichfeldt and showed that if \(\chi\) is a generalized character of a finite group \(G\) and \(L=\{\chi(g)\mid 1\neq g\in G\}\), then \(f_L(\chi)=m\rho\) where \(\rho\) is the regular representation. \(\chi\) is called sharp if \(m=1\). The authors investigate these concepts and relate them to the Burnside ring. Recall that the mark of a subgroup acting on a \(G\)-set is the number of fixed points of that subgroup. In Theorem 4.1, they show that certain sets of marks do come from actual \(G\)-sets. They also give a different proof of the known result that if a faithful \(G\)-set \(X\) has \(r\) distinct marks, then \(X\) has a base of size \(r\) (i.e. a subset of size \(r\) such that no element of \(G\) fixes every element of the subset). The authors give some interesting open questions.