Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / B / 8 / a / Solution

The simple modules are indexed by the p-regular partitions $\lambda$ of $n$, meaning no part is repeated $p$ or more times. Equip the tabloid permutation module with the bilinear form invariant under the symmetric group making its tabloid basis orthonormal. The construction isThe standard modular Specht theorem says this quotient is nonzero precisely for p-regular $\lambda$, is then simple, and these quotients give all simple modules.

To verify distinctness of labels, use the standard dominance triangularity theorem: if $D^{\nu }$ occurs as a composition factor of $S^{\lambda }$, then $\nu \ge \lambda$; for a p-regular label the diagonal multiplicity is one. If $D^{\lambda }\cong D^{\mu }$, then $D^{\lambda }$ occurs in $S^{\mu }$, so $\lambda \ge \mu$, and the symmetric argument gives $\mu \ge \lambda$. Antisymmetry of the dominance order on partitions yields $\boxed{\lambda =\mu }$. These are the general representation-theoretic results allowed in this part.