On the \(\mathcal S_{n}\)-modules generated by partitions of a given shape (Q1010843)

Summary: Given a Young diagram \(\lambda\) and the set \(H^{\lambda}\) of partitions of \(\{1,2,\dots, |\lambda|\}\) of shape \(\lambda\), we analyze a particular \({\mathcal S}_{|\lambda|}\)-module homomorphism \({\mathbb Q}H^{\lambda}\to{\mathbb Q}H^{\lambda'}\) to show that \({\mathbb Q}H^{\lambda}\) is a submodule of \({\mathbb Q}H^{\lambda'}\) whenever \(\lambda\) is a hook \((n,1,1,\dots,1)\) with \(m\) rows, \(n\geq m\), or any diagram with two rows.