The projective characters of the symmetric groups that remain irreducible on subgroups (Q1174743)

The main problem solved here is to determine completely the set of maximal subgroups of the symmetric group \(S_ n\) for which an irreducible spin character \(\zeta^ \lambda\) (where \(\lambda\) is a strict partition) of \(S_ n\) remains irreducible on restriction to the subgroup. In fact, it is shown that these subgroups are (i) \(A_ n\) and \(\lambda\) odd, (ii) \(S_{n-1,1}\), \(\lambda = \{\ell + r,\ell + r - 1,\dots, r + 1\}\), \(n = \textstyle{1\over 2}\ell(\ell + 1) + r\ell\), \(\ell \geq 2\) and with \(\lambda\) odd and \(r \geq 0\) or with \(\lambda\) even and \(r = 0\), (iii) \(S_{n-2,2}\), \(\lambda = \{\ell,\ell - 1,\dots,1\}\), \(n = \textstyle{1\over 2} \ell(\ell + 1)\), (iv) \(S_{n- a,a}\), \(\lambda = \{n\}\), \(n\) even, \(a < \textstyle{1\over 2}n\), (v) \(S_ a \wr S_ b\), \(n = ab\), \(a,b \geq 2\), \(\lambda = \{n\}\), (vi) \(S_ 5 \wr S_ 2\), \(n = 10\), \(\lambda = \{4,3,2,1\}\), (vii) \(S_ 2 \wr S_ 3\) or \(S_ 3 \wr S_ 2\), \(n = 6\), \(\lambda = \{3,2,1\}\), (vii) \(\mathbb{Z}_ 5:\mathbb{Z}_ 4\), \(n = 5\), \(\lambda = \{5\}\) or \(\{32\}\), (ix) \(S_ 5\), \(n = 6\), \(\lambda = \{6\}\) or \(\{3,2,1\}\), (x) \(\text{Aut}(A_ 6)\), \(n = 10\), \(\lambda = \{10\}\), (xi) \(M_{12}\), \(n = 12\), \(\lambda = \{12\}\). Similar results are obtained for the alternating groups and also for quasisimple subgroups of the alternating groups. Needless to say, the proof involves a detailed and meticulous case by case investigation.