  By exploiting relationships between the values taken by ordinary characters
of symmetric groups we prove two theorems in the modular representation theory
of the symmetric group.
  1. The decomposition matrices of symmetric groups in odd characteristic have
distinct rows. In characteristic 2 the rows of a decomposition matrix labelled
by the different partitions $\lambda$ and $\mu$ are equal if and only if
$\lambda$ and $\mu$ are conjugate. An analogous result is proved for Hecke
algebras.
  2. A Specht module for the symmetric group $S_n$, defined over an
algebraically closed field of odd characteristic, is decomposable on
restriction to the alternating group $A_n$ if and only if it is simple, and the
labelling partition is self-conjugate. This result is generalised to an
arbitrary field of odd characteristic.
