The depth of Young subgroups of symmetric groups. (Q375194)

For a subgroup \(H\) of a finite group \(G\), the depth of the inclusion \(H\leqslant G\) is the minimum depth of the inclusion of the group algebras \(\mathbb CH\subseteq\mathbb CG\). Let \(n\in\mathbb N\) and take \(G=\mathfrak S_n\), the symmetric group of degree \(n\). For appropriate values of \(k\in\mathbb N\) there exist Young subgroups of \(G\) of the form \(Y_1=\mathfrak S_{n-k}\times\mathfrak S_1\times\cdots\times\mathfrak S_1\cong\mathfrak S_{n-k}\) and \(Y_2=\mathfrak S_{n-k}\times\mathfrak S_k\). Using combinatorial methods from the representation theory of the symmetric group, and exploiting various equivalent notions of depth, the authors compute the depth of \(Y_1\) and \(Y_2\) in \(G\). One important general method deployed in the proof is the use of base sizes of permutation groups to obtain an upper bound for the depth of subgroups.NEWLINENEWLINE Namely, in detail, for \(1\leqslant k<n\), \(Y_1\) has depth \(2\lfloor(n-2)/k\rfloor+1\) in \(G\). Next, assume \(n\geqslant 2k\). Then the depth of \(Y_2\) in \(G\) is \(2\lceil 2(n-1)/(k+1)\rceil-1\) (if \(n\geqslant (k^2+2)/2\)), and it is \(2\lceil\log_2k\rceil+1\) (if \(n=2k\)). Upper and lower bounds for the depth are given in the case when \(2k<n<(k^2+2)/2\), although precise values are also determined for all \(k\leqslant 10\) and \(n\leqslant 39\).