Multiplicity-free induced characters of symmetric groups (Q6631318)

Let \(G\) be a finite group. An ordinary character \(\chi\) of \(G\) is multiplicity free if the irreducible constituents of \(\chi\) appear with multiplicity one. \textit{M.Wildon} [J. Pure Appl. Algebra 213, No. 7, 1464--1477 (2009; Zbl 1170.20006)] and independently \textit{C. Godsil} and \textit{K. Meagher} [Ann. Comb. 13, No. 4, 463--490 (2010; Zbl 1234.20015)] classified all multiplicity-free permutation characters of the symmetric group \(S_{n}\) for \(n \geq 66\).\N\NIn the paper under review, the author investigates for which pairs \((G,\rho)\) of a subgroup \(G\) of the symmetric group \(S_{n}\) and \(\rho \in G\) the induced character \(\rho \uparrow^{S_{n}}\) is multiplicity-free. As a result, for \(n \geq 66\), he classifies all subgroups \(G \leq S_{n}\) that yield such a pair. In addition, for many of these groups \(G\) he identifies all the possible choices of the irreducible character \(\rho\) assuming \(n \geq 73\).