Branching rules for Specht modules

Let Sλ be a Specht module for the symmetric group Σn, defined over a field of characteristic different from 2, and let Ln−1 be the sum of all transpositions in Σn−1 that do not fix n−1. It is shown that the minimal polynomial of Ln−1 acting on Sλ has maximum possible degree. As a consequence, the indecomposable components of the restriction of Sλ to Σn−1 coincide with the block components. Analogous results are proved for Ln+1 and the Σn+1-module that is induced from Sλ.